Process of polymerizing tetra-functional long-chain branched polyolefin resins

ABSTRACT

The present process embodiments for synthesizing long-chain branched copolymers include contacting together one or more C 2 -C 14  alkene monomers, at least one diene or polyene, optionally a solvent, and a multi-chain catalyst. The multi-chain catalyst includes a plurality of polymerization sites and produces at least two polymer chains of the C 2 -C 14  alkene monomers, each polymer chain polymerizing at one of the polymerization sites. The process synthesizes the long-chain branched polymers by connecting the two polymer chains with the diene or polyene, the joining of the two polymer chains being performed in a concerted manner during the polymerization.

CROSS REFERENCE TO RELATED APPLICATION

This application claims priority to U.S. Provisional Patent ApplicationNo. 62/738,621 filed on Sep. 28, 2018, the entire disclosure of which ishereby incorporated by reference.

TECHNICAL FIELD

Embodiments of the present disclosure generally relate to polymercompositions having long-chain branches and the process by which thepolymer compositions are synthesized.

BACKGROUND

Olefin based polymers, such as polyethylene and polypropylene, areproduced via various catalyst systems. Selection of such catalystsystems used in the polymerization process of the olefin based polymersis an important factor contributing to the characteristics andproperties of such olefin based polymers.

Polyethylene and polypropylene are manufactured for a wide variety ofarticles. The polyethylene and polypropylene polymerization process canbe varied in a number of respects to produce a wide variety of resultantpolyethylene resins having different physical properties that render thevarious resins suitable for use in different applications. The amount ofshort-chain branching in a polyolefin affects the physical properties ofthat polyolefin. The effect of branching on properties of polyethylenedepends on the length and the amount of branches. Short branches mainlyinfluence the mechanical and thermal properties. As the short-chainbranching frequency increases, the polymer is less able to form lamellarcrystals, and the mechanical and thermal properties diminish. Smallamounts of long-chain branching can alter the polymer processingproperties significantly.

To form long-chain branching, a vinyl or terminal double bond of apolymer chain is incorporated into a new polymer chain. Reincorporationof vinyl terminated polymers and introducing a diene comonomer are twomechanisms by which a vinyl group on a polymer strand is incorporatedinto a second polymer strand. Additionally, long-chain branching isinduced via radicals. It is difficult to control the amount of branchingin all three mechanisms. When using radicals or dienes to initiatelong-chain branching, the branching may become too numerous, therebycausing gelling and reactor fouling. The reincorporation mechanism doesnot produce much branching, and branching can only occur after thepolymer strand is produced, thereby further limiting the amount ofbranching that can occur.

SUMMARY

Embodiments of this disclosure include processes for synthesizinglong-chain branched copolymers. The process includes contacting togetherone or more C₂-C₁₄ alkene monomers, at least one diene or polyene,optionally a solvent, and a multi-chain catalyst. The multi-chaincatalyst includes a plurality of polymerization sites and produces atleast two polymer chains of the C₂-C₁₄ alkene monomers, each polymerchain polymerizing at one of the polymerization sites. The processsynthesizes the long-chain branched polymers by connecting the twopolymer chains with the diene or polyene, the joining of the two polymerchains being performed in a concerted manner during the polymerization.

Various embodiments of the process include polymerizations that occur ina solution polymerization reactor or a particle forming polymerizationreactor such as a slurry reactor or a gas phase reactor, wherein themolecular or solid-supported catalyst is delivered to the reaction mediaor developed in the reaction media, wherein the reactor system is batchor continuous or a hybrid such as semi-batch, wherein the reactorresidence time distribution is narrow such as in non-backmixed reactorsor broad such as in backmixed reactor and series and recycle reactors.

BRIEF DESCRIPTION OF FIGURES

FIG. 1 is a graphical depiction of the molecular weight of a polymer asthe number of branching methines per 1000 carbons increase.

FIG. 2 is a graphical model predicted dependence of molecular weightdistribution (MWD) curve on branching level.

FIG. 3 is a graphical model predicted dependence of relative peakmolecular weight on branching level.

FIG. 4 is a graphical depiction of a predicted dependence of themolecular weight distribution (MWD) curve on tri-functional dienesbranching level.

FIG. 5 is a graphical depiction of a predicted dependence of relativepeak of the molecular weight (MW) on tri-functional dienes branchinglevel.

FIG. 6 is a graphical depiction of a model-predicted effect of branchingon peak molecular weight (M_(p)) versus branches per polymer molecule ofconventional diene branching (solid line) and “Ladder branching” (dashedline).

FIG. 7 is a graphical depiction of a model-predicted effect of branchingon weight average molecular weight (M_(w)) versus branches per polymermolecule of conventional diene branching (solid line) and “LadderBranching” (dashed line).

FIG. 8 is a graphical depiction of a model-predicted effect of branchingon peak weight average molecular weights (M_(p)) versus branches perlinear chain segment of conventional diene branching (solid line) (vs.B_(c)) and the M_(p) of “Ladder Branched” polymers (dashed line) (vs.R_(c)).

FIG. 9 is a graphical depiction of a model-predicted effect of branchingweight average molecular weight (M_(w)) versus branches per linear chainsegment for conventional diene branching (vs. B_(c)) (solid line) and“Ladder Branched” polymer (dashed line) (vs. R_(c)).

FIG. 10 is a graphical depiction of MWD slopes used to calculate shapemetrics G(79/29) and G(96/08) where S(X) is the slope at X % of the MWDheight. G(A/B)=(S(A)−S(B))/S(A).

FIG. 11 is a graphical depiction of a model predicted molecular weightdistribution (MWD) shape metric G(79/29) compared for conventional and“Ladder Branching” as a function of branching level as depicted byrelative peak MW(M_(p)/M_(po)).

FIG. 12 is a graphical depiction of a model predicted MWD shape metricG(79/29) compared for conventional and “Ladder Branching” as a functionof branching level as depicted by relative weight average MW(M_(w)/M_(wo)).

FIG. 13 is a graphical depiction of a model predicted MWD shape metricG(98/08) compared for conventional and “Ladder Branching” as a functionof branching level as depicted by relative peak MW (M_(p)/M_(po)).

FIG. 14 is a graphical depiction of a model predicted MWD shape metricG(98/08) compared for conventional and “Ladder Branching” as a functionof branching level as depicted by relative weight average MW(M_(w)/M_(po)).

FIG. 15 is a graphical depiction of the MWD curve illustrating how thehigh MWD tail area metrics are defined using the point of maximum slope.

FIG. 16 is a model predicted MWD area metric, A_(HIGH), compared forconventional and “Ladder Branching” as a function of branching level asdepicted by relative weight average molecular weight (M_(p)/M_(po)).

FIG. 17 is a model predicted MWD area metric, A_(HIGH), compared forconventional and “Ladder Branching” as a function of branching level asdepicted by relative peak molecular weight (M_(w)/M_(po)).

FIG. 18 is a model predicted MWD area metric, A_(TAIL), compared forconventional and “Ladder Branching” as a function of branching level asdepicted by relative peak molecular weight (M_(p)/M_(po)).

FIG. 19 is a model predicted MWD area metric, A_(TAIL), compared forconventional and “Ladder Branching” as a function of branching level asdepicted by relative weight average molecular weight (M_(w)/M_(po)).

FIG. 20 is a graph of the absolute molecular weight distribution (MWD)curve as measured by GPC for Example Series 2.4 as recorded in Table 2.

FIG. 21 is a conventional molecular weight distribution curve measuredby conventional gel permeation chromatography (GPC).

FIG. 22 is an absolute molecular weight distribution curve measured byGPC triple light scattering detector (also called Absolute GPC).

FIG. 23 is a graph of the extensional viscosity as measured as afunction of time in seconds for a “Ladder Branched” polymer resin.

FIG. 24 is a graph of the melt strength (cN) as a function of velocity(mm/s) for a “Ladder Branched” polymer resins.

FIG. 25 is a conventional molecular weight distribution curve measuredby GPC of an unbranched ethylene-based polymer and a “Ladder Branched”polymer resin.

FIG. 26 is an absolute molecular weight distribution curve measure by aGPC triple light scattering detector of an unbranched ethylene-basedpolymer and a “Ladder Branched” polymer resin.

FIG. 27 is a graph of the extensional viscosity as measured as afunction of time in seconds of a “Ladder Branched” polymer resin.

FIG. 28 is a graph of the melt strength (cN) as a function of velocity(mm/s) of a “Ladder Branched” polymer resin.

FIG. 29 is an absolute molecular weight distribution curve measured by aGPC triple light scattering detector for two comparative examples havingno diene and four samples having a variable amount of diene.

FIG. 30A is a graph of the absolute molecular weight distributions ofcomparative conventional branched polymer samples with varying amountsof diene.

FIG. 30B is a graph of the conventional molecular weight distributionsof comparative conventional branched polymer samples with varyingamounts of diene.

FIG. 31 is a graph of the rheology ratio as a function of average g′ forvarious polymer resins and “Ladder Branched” polymer resins.

FIG. 32 is a graph of the rheology ratio as a function of polydispersityindex (PDI) for various polymer resins and “Ladder Branched” polymerresins.

FIG. 33 is a graph of the melt strength (centiNewtons, cN) as a functionof the melt index (Log I₂) of polymers produced with a single chaincatalyst and a dual chain catalyst, with additional lines depictinglinear polyethylene, tubular low density polyethylene and autoclave lowdensity polyethylene.

DETAILED DESCRIPTION

Specific embodiments of a process for synthesizing polymer and polymerssynthesized by the process of this disclosure will now be described. Itshould be understood that the process for synthesizing polymers of thisdisclosure may be embodied in different forms and should not beconstrued as limited to the specific embodiments set forth in thisdisclosure. Rather, embodiments are provided so that this disclosurewill be thorough and complete, and will fully convey the scope of thesubject matter to those skilled in the art.

Definitions

The term “polymer” refers to a polymeric compound prepared bypolymerizing monomers, whether of the same or a different type. Thegeneric term polymer thus embraces the term “homopolymer,” usuallyemployed to refer to polymers prepared from only one type of monomer aswell as “copolymer” which refers to polymers prepared from two or moredifferent monomers. The term “interpolymer,” as used herein, refers to apolymer prepared by the polymerization of at least two different typesof monomers. The generic term interpolymer thus includes copolymers, andpolymers prepared from more than two different types of monomers, suchas terpolymers.

“Polyethylene” or “ethylene-based polymer” shall mean polymerscomprising greater than 50% by weight of units which have been derivedfrom ethylene monomer. This includes polyethylene homopolymers orcopolymers (meaning units derived from two or more comonomers). Commonforms of polyethylene known in the art include Low Density Polyethylene(LDPE); Linear Low Density Polyethylene (LLDPE); Ultra Low DensityPolyethylene (ULDPE); Very Low Density Polyethylene (VLDPE); single-sitecatalyzed Linear Low Density Polyethylene, including both linear andsubstantially linear low density resins (m-LLDPE); Medium DensityPolyethylene (MDPE); and High Density Polyethylene (HDPE).

Embodiments of this disclosure include a process of synthesizinglong-chain branched polymers by adding a C₂ monomer, optionally at leastone or more C₃-C₁₂ α-olefin comonomer, at least one diene, a multi-chaincatalyst, and optionally, a solvent, in which the multi-chain catalystincludes a molecule having a plurality of polymerization sites,producing at least two copolymer strands, each copolymer strandcopolymerizing at one of the polymerization sites; and synthesizing thelong-chain branched polymers by connecting the two copolymer strandswith the diene, the connection of the two copolymer strands beingperformed concertedly with the copolymerization.

The process of synthesizing polymers according to this disclosure isdifferent from the conventional long-chain branching. The term“long-chain branching” refers to branches having greater than 100 carbonatoms. A “branch” refers to a portion of polymer that extends from atertiary or quaternary carbon atom. When the branch extends from atertiary carbon atom, there are two other branches, which collectivelycould be the polymer strand from which the branch extends.Conventionally, long-chain branching (LCB) may occur naturally in thepolymerization process, as shown in Scheme 1. This can occur throughvinyl termination of the polymer chain and reinsertion of themacromolecular vinyl creating a tri-functional long-chain branch.Depending on the degree of branching, a variety of methods can eitherdetermine LCB, such as nuclear magnetic resonance (NMR), or distinguishthe effect of LCB in the polymer. For example, the effect of LCB isobserved in shear flow in the van Gurp-Palmen analysis, also an increaseof the shear viscosity at low angular frequencies and strength of theshear thinning behavior can be attributed to LCB. In extensional flow,the influence of LCB is usually identified in the degree of hardening orthe strength of the melt and the maximum deformation achieved. A highlevel of natural LCB in a polymer is difficult to achieve due to thelimited concentration of vinyl terminated polymers (maximum one perpolymer chain) and the need to run to high ethylene conversion to ensureLCB formation. To ensure high conversion, there is a low ethyleneconcentration in the reactor, thus enabling a great amount of vinylterminated polymers to be reinserted in a second polymer chain.

In Scheme 1, “Cat” is the catalyst and “P” is the polymer chain.

There is minimal long-chain branching that forms through the naturallyoccurring branching process. One way to enhance LCB is through theaddition of α,ω-dienes to the polymerization system, whether it be in aradical, heterogeneous, or homogeneous process. In general, dienes addto the polymer chain in a similar manner to α-olefins, but leave apendant vinyl group which can insert into a polymer chain a second timeto create the LCB, as illustrated by Scheme 2. In general, the dienelength does not matter, only that it can link two polymer chainstogether. In principle, the concentration of pendant vinyls can becontrolled through the amount of diene added to the reactor. Thus, thedegree of LCB can be controlled by the concentration of pendant vinyls.

In Scheme 2, “Cat” is the catalyst; “P” is the polymer chain; and thediene in this example is 1,5-hexadiene.

The conventional process of incorporating dienes into a polymersynthesis system suffers from the fundamental flaw of gel formation orreactor fouling. The kinetic modeling, discussed in later paragraphs,may provide good predictive results that enable a better understandinginto gel formation. For example, longer polymer chains have moreinserted olefins, thus more inserted dienes, thus more pendant vinyls,implying that longer polymer chains will be more likely to re-insertinto the catalyst to form a LCB. Thus, the longer polymer chainspreferentially re-insert forming tetra-functional branches, which areeven larger polymer molecules, and a gel problem results. As indicatedin Scheme 2, a tetra-functional LCB has a short segment (number ofcarbons between the two double bonds of the diene), which bridges twolong chains on each side of the short segment. A simulation of theweight average molecular weight (M_(w)) and number average molecularweight (M_(n)) as a function of branching is shown in FIG. 1 forpolyethylene in a semi-batch reactor at constant pressure. In FIG. 1,M_(n) only marginally increases as M_(w) becomes infinite. As the M_(w)increases to a number greater than 200,000 grams per mole (g/mol), thepolymer gels, gelling occurs, or reactor fouling is present.

The term “gel” or “gelling” refers to a solid composed of at least twocomponents: the first is a three dimensional cross-linked polymer andthe second is a medium in which the polymer does not fully dissolve.When the polymer gels and does not fully dissolve, the reactor maybecome fouled with polymer gel.

The term “Ladder Branched” polymer refers to the tetra-functional longchain branched polymer as disclosed in this application, and the term or“Ladder Branching mechanism” refers to how the “Ladder Branched”polymers are formed.

In one or more embodiments of this disclosure, the process to synthesizethe long-chain branched polymer achieves long-chain branching and avoidsgel formation or reactor fouling. Without intending to be bound bytheory, it is believed that reactor fouling is avoided by reacting thetwo alkenes of the diene in a concerted fashion across two proximalpolymer chains. For example and illustrated by Scheme 3, one alkene ofthe diene reacts before the second alkene, and the second alkene reactsbefore too many ethylene molecules are added to the polymer strand,thereby removing the close proximity the second alkene has to thereactive site. The reaction of the first alkene of the diene into onepolymer and second alkene of the diene into an adjacent polymer chainbefore many ethylene monomers are inserted is referred to as a concertedaddition of the diene into proximal polymer chains.

Polymer strands are linear segments of a polymer, or more specifically acopolymer, which are optionally joined at the end(s) by branchingjunctures. For example, a tetra-functional branch juncture joins theends of four polymer strands, as opposed to a tri-functional branchjuncture, which joins the ends of three polymer strands as shown inScheme 1.

The combination of a multi-chain catalyst and diene influences theamount and type of branching. Embodiments of the present disclosure aredirected to controlling polymer properties such as: 1) the use ofmultiple diene species, 2) the use of multiple multi-chain catalystspecies, or 3) the combination of polymerization environments includingmultiple reactors zones or a gradient of zones.

Although, using multiple catalysts, including single-chain catalysts,may allow conventional branching. The use of multiple dienes speciesalso includes those dienes which do not create branches or tend toresult in “conventional” LCB. The process of synthesizing polymersaccording to this disclosure is different from the conventionallong-chain branching. The term “long-chain branching” refers to brancheshaving greater than 100 carbon atoms. The term “branch” refers to aportion of polymer that extends from a tertiary or quaternary carbonatom. When the branch extends from a tertiary carbon atom, there are twoother branches, which collectively could be the polymer chain from whichthe branch extends. Long-chain branching (LCB) may occur naturally inthe polymerization process, as shown in Scheme 1. This may occur throughtermination of the polymer chain and reinsertion of the macromolecularvinyl creating a tri-functional long-chain branch.

In one or more embodiments, the process for polymerizing the long-chainbranched polymer includes a catalyst with at least two active sites inclose proximity (multi-chain catalysts). In order for the two activesites to be in close proximity, the two active sites may be less than18.5 angstroms (Å) apart. In some embodiments, the two active sitesinclude a distance of from 2.5 angstroms (Å) to 18.5 Å, from 9 Å to 14Å, or approximately 11 Å. In various embodiments, the process forpolymerizing the long-chain branched polymer includes a multi-chaincatalyst. In one or more embodiments, the multi-chain catalyst mayinclude at least one metal center, in which the two active site are onthe same metal center. In some embodiments, the multi-chain catalyst mayinclude a metal-ligand complex, in which the two active sites (twopolymer chains) are on the same metal center.

According to an X-ray crystal structure (A. D. Bond, Chem. Comm. 2002,1664), 1,9-decadiene has a distance between terminal carbons of 10.8 Å.While there is data that the 1,9-decadiene form rungs between twopolymer chains via the “Ladder Branching” mechanism, one may believethat α,ω-dienes having more than 10 carbon atoms may also form rungs viathe “Ladder Branching” mechanism. Without intent to be bound by theory,the issue of whether α,ω-dienes having more than 10 carbon atoms mayform rungs may be determined by the distance between the two polymerchains. For example, when the two polymer chains reside on differentmetal atoms of a catalyst (e.g., bimetallic, heterogeneous), theα,ω-dienes may include additional methylene units (same C—C bond lengthsand angles) to extend this structure to 1,15-hexadecadiene. Withoutintent to be bound by theory, it is presumed this 16-carbon analog stillhas the potential to form a rung via the “Ladder Branching” mechanism.In this manner, one can consider dienes, 1,11-dodecadiene (13.3 Ådistance between terminal carbons), 1,13-tetradecadiene (15.9 Å distancebetween terminal carbons), 1,15-hexadecadiene (18.5 Å distance betweenterminal carbons). In some embodiments, when the dual chain catalyst inthe “Ladder Branching” mechanism is a bimetallic catalysts, the diene isless than or equal to 18.5 Å.

It is well-known that modem computational techniques can reproduce knownexperimental crystal structures to high accuracy as a way to estimatedistances between chains for a catalyst. For a heterogeneous system, onemay estimate surface concentration of metals which are often measured inmetal atoms per nanometer squared (M/nm²). This surface coverageprovides an estimate of accessible metals on the surface which if evenlydispersed may be converted to an M-M distance, which reflects thedistance between the polymer chains. For an extended surface, 1metal/nm² leads to a distance of 10 Å between the metal atoms, wellwithin the desired cut-off. At 18.5 Å, one can determine the coverage at0.3 metal/nm².

Examples of catalysts having at least two active sites, wherein theactive sites are in close proximity include, but are not limited to,bimetallic transition metal catalysts; heterogeneous catalysts;dianionic activators with two associated active catalysts; a ligatedtransition metal catalyst with more than one propagating polymer chain;a group IV olefin polymerization catalyst including monoanionic groups,bidentate monoanionic groups, tridentate monoanionic groups, or amonodentate, bidentate, or tridentate monoanionic groups with externaldonors.

The catalysts in Table 1 are illustrative embodiments of the classes ofcatalysts previously described and specific catalysts contemplated. Theexamples in Table 1 are not intended to be limiting; rather they aremerely illustrative and specific examples for the classes of catalystpreviously mentioned.

TABLE 1 Catalysts with more than one active site in close proximityClass Illustrative Specific Bimetallic Catalysts

Heterogeneous and Supported Catalysts

Group IV Olefin Polymerization Catalyst

While not intending to be bound by theory, a mechanism, as explained inthis section, describes how a dual chain catalyst can create a uniquebridged molecular architecture when polymerizing diene co-monomers underdesired conditions. The term “diene” refers to a monomer or moleculehaving two alkenes. A pictorial description of the kinetics is shown inScheme 4, in which the catalyst center produces two polyolefin chains.Scheme 4 shows how a combination of diene bridging and chain transfermay create a diene “Ladder Branched” polymer structure. The term diene“Ladder Branched” polymer refers to the long-chain branching, in which ashort chain or rung that includes one to twelve carbon atoms links twolong-chains together. As shown, the metal-ligand catalyst having atleast two polymer chain sites propagates two separate polymer chains.One alkene of the diene is incorporated into one of the sites of thecatalyst, and it is believed that due to the close proximity of thepropagation sites, the second alkene of the diene is then quicklyincorporated into the second polymer chain, thereby forming a bridge orrung. This successive addition of diene is referred to as a “concerted”addition of the diene, distinguishing it from catalysts without twoproximal chains where diene addition leads to a concentration of vinylcontaining polymers in the reactor, which react at a later time. Theterm “rung” refers to the diene once it is incorporated into twoseparate polymer strands, thereby linking the strands together. Thefirst and second polymer strands continue to propagate until the polymertransfers to another catalyst, the polymer is released from thecatalyst, the catalyst dies, or another diene is added.

Kinetics

Without intending to be bound by theory, it is believed that themolecular weight distribution associated with these proposed kinetics isinherently stable at high branching levels when the diene bridgingreaction is the sole source of branching. The molecular weightdistribution (MWD) is defined by the weight average molecular weightdivided by the number average molecular weight (M_(w)/M_(n)). Theinherent stability of the MWD means that the weight average molecularweight (M_(w)) increases only moderately even at high branching levels,which is in contrast to conventional diene comonomer branchingtechnology wherein M_(w) and M_(w)/M_(n) become infinite at moderatetetra-functional branching levels.

A mathematical model is derived for the purpose of demonstrating how theprocess for synthesizing polyethylene creates a long-chain branchedpolymer having a diene “Ladder Branched” molecular architecture. Themathematical model will also be used to establish claims metrics andranges. The mathematical model of the branched architecture as describedin this disclosure may be derived from a kinetics description of theproposed mechanism of branching. This model is based upon severalassumptions to facilitate mathematical simplicity, but these assumptionsare not intended to limit the scope of this disclosure. The assumptionsfollow common industrial applications of non-living addition ofcopolymers as well as additional assumptions specific to the assumeddiene branching mechanism. The common assumptions made include: (1)propagation is much faster than chain transfer, therefore average chainlength is much greater than one monomer; (2) only a single pure catalystspecies is active; (3) the catalyst center makes many chains during itslifetime, and therefore the chain lifetime is a small fraction of thereaction or residence time; (4) co-polymerization may be approximated bya homopolymerization model when there is negligible composition drift.

Kinetics for Diene “Ladder Branching” Theory

In addition to the four commonly made assumptions, there are fourassumptions, on which the kinetics for diene “Ladder Branching” theoryis based. The first assumption is that the catalyst centersimultaneously produces two kinetic chains with identical kinetics andstatistics. Secondly, the rung is formed when a diene bridges twopolymer chains increasing in length. Third, the branch point is formedwhenever two un-bridged chains are bridged by a diene. Finally, thediene reactions not forming bridges are ignored since the MWD is notaffected.

The kinetic description of a proposed diene “Ladder Branching” mechanismrequires the deployment of a nomenclature that describes how eachreaction affects the molecular architecture. Some nomenclature elementsbelow represent small molecules (M, A, D) while the other nomenclatureelements represent the molecular architecture (P_(n,m), S_(n), D_(n)).The kinetics will show how the nomenclature elements interact to formthe molecular architecture.

Kinetic Nomenclature

M: monomer or co-monomers; A: chain transfer agent species; D: dienebranching species; n, m: indices reflecting the number of monomericrepeat units on a subspecies; P_(n,m): catalyst with two un-bridgedpropagating polymers having n and m monomeric repeat units; D_(n): deadpolymer molecular with n monomeric repeat units; S_(n): catalystproducing a bridged polymer molecule with n monomeric repeat units; Kc:kinetic chains are defined as linear segments created by chain transfer;Rg: rungs are defined as bridges between chain segments; Br: branchesare created when two previously un-bridged molecules become bridged.

The equations for branching kinetics are written below using thenomenclature and assumptions introduced above. A brief description willbe given for each reaction, and anyone skilled in the art ofpolymerization kinetics should be capable of comprehending the kineticscheme and rate laws.

TABLE 2 Kinetics for Diene “Ladder Branching” Theory, (n ≥ l, m ≥ 1)Chain Reaction position Reaction Constant Propagation (left) P_(n,m) + M→ P_(n+1),_(m) k_(p), (L/mole/sec) (right) P_(n,m) + M → P_(n,m+1)k_(p), (L/mole/sec) S_(n) + M → S_(n+1) 2 k_(p), (L/mole/sec) ChainTransfer (left) P_(n,m) + A → P_(0,m) + D_(n) + kc k_(tra), (L/mole/sec)(right) P_(n,m) + A → P_(n,0) + D_(m) + kc k_(tra), (L/mole/sec) (right)P_(n,m) + A → P_(n,0) + D_(m) + kc k_(tra), (L/mole/sec) (left) S_(n) +A → P_(0,n) + kc k_(tra), (L/mole/sec) (right) S_(n) + A → P_(n,0) + kck_(tra), (L/mole/sec) Diene Bridging P_(n,m) + D → S_(n+m) + br + rg 4k_(d), (L/mole/sec) S_(n) + D → S_(n) + rg 4 k_(d), (L/mole/sec)Re-initiation P_(n,0) + M → P_(n,1) fast reaction P_(0,m) + M → P_(1,m)fast reaction

The outcome of propagation is the incremental increase in chain size byone repeat unit. Propagation is written separately for each of the twomolecules increasing in length from a catalyst center. For example, thefirst index on P_(n,m) is for the left chain on the catalyst and thesecond index is for the right chain on the catalyst. When propagation ismodeled for an increase in length bridged molecule (S_(n)), a factor of2 appears in the rate, because there are two chain positions, the leftand the right, on each center equally available for reaction.

Chain transfer, like propagation, is written separately for the left andright positions on the catalyst. Chain transfer of an un-bridged species(P_(n,m)) produces a dead polymer molecule (D_(n) or D_(m)) and a vacantposition (P_(0,m) or P_(n,0)). When a propagating bridged molecule(S_(n)) engages in chain transfer, an un-bridged species (P_(n,0) orP_(0,n)) is produced and no dead polymer is produced since all n repeatunits are still bonded to the catalyst. The vacant positions (P_(0,m)and P_(n,0)) resulting from chain transfer are assumed to very quicklyre-initiate and engage in propagation. The rate expressions for dienebridging include a factor of 4 because each diene has two polymerizablegroups and each catalyst center has two positions (left and right) fordiene incorporation.

Diene bridging results in the formation of a tetra-functional branch(br) only when un-bridged (P_(n,m)) species react productively with adiene. A tetra-functional branch refers to a short segment where fourpolymer chains can emanate from, two from each side of the shortsegment. With dienes, tetra-functional branches are the expected type ofLCB. A rung is produced (rg) when any catalyst center productivelyincorporates a diene regardless of whether it has bridged (S_(n)) orun-bridged (P_(n,m)) molecules. Diene reactions that do not result inbridging, such as intra-chain cyclization and pendant vinyl formation,are ignored and are considered non-productive for these kinetics.

The creation of a model from the kinetics requires that a series ofpopulation balances be derived for each type of polymer speciesinvolved. These population balances are derived as a function of chainlength (n, m) and represent kinetic rates of change of the variouspolymeric subspecies. The population balances are given below assumingmass action rate laws with P_(n,m), S_(n), and D_(n) symbolsrepresenting the molar concentration of subspecies for n≥1 and m≥1. Thekinetics model can be extended to include other chain transferreactions, such as with hydrogen (k_(trh)) and beta hydride elimination(k_(b)) merely by expanding the definition of the transfer term,Ω=k_(tra)A+k_(trh)H₂+k_(b).

$\begin{matrix}{\mspace{79mu}{R_{S_{n}} = {{2\Psi{\sum\limits_{m = 1}^{n - 1}\left( {P_{m,{n - m}} + P_{{n - m},m}} \right)}} + {2{\Phi\left( {S_{n - 1} - S_{n}} \right)}} - {2\Omega S_{n}}}}} & (1) \\{R_{p_{n,m}} = {{\Phi\left( {P_{{n - 1},m} + P_{n,{m - 1}} - {2P_{n,m}}} \right)} - {\left( {{2\Omega} + {4\Psi}} \right)P_{n,m}} + {\delta_{m - 1}{\Omega\left( {S_{n} + L_{n}} \right)}} + {\delta_{n - 1}{\Omega\left( {S_{m} + R_{m}} \right)}}}} & (2) \\{\mspace{79mu}{R_{D_{n}} = {{\Omega{\sum\limits_{s = 1}^{\infty}\left( {P_{s,n} + P_{n,s}} \right)}} = {\Omega\left( {R_{n} + L_{n}} \right)}}}} & (3)\end{matrix}$

In equations (1), (2), and (3):

$\begin{matrix}{\Omega = {k_{tra}A}} & (4) \\{\Psi = {k_{d}D}} & (5) \\{\Phi = {k_{p}M}} & (6) \\{L_{n} = {\sum\limits_{s = 1}^{\infty}P_{n,s}}} & (7) \\{R_{n} = {{\sum\limits_{s = 1}^{\infty}{P_{s,n}\mspace{31mu}\delta_{i}}} = \left\{ \begin{matrix}{0,{i \neq 0}} \\{1,{i = 0}}\end{matrix} \right.}} & (8)\end{matrix}$

Other important population balances may be derived from equations (1) to(8), such as the propagating polymer subspecies distributions for theleft side (L_(n)) and right side (R_(n)). The left and right sidedistributions of the propagating polymer subspecies are equal, due tosymmetry imposed in defining the kinetics scheme.

R _(L) _(n) =Φ(L _(n-1) −L _(n))−(Ω+4Ψ)L _(n) +ΩS_(n)+δ_(n-1)Ω(ξ_(0,0)+μ₀)  (9)

R _(R) _(n) =Φ(R _(n-1) −R _(n))−(Ω+4Ψ)R _(n) +ΩS_(n)+δ_(n-1)Ω(ξ_(0,0)+μ₀)  (10)

The rates of formation of molecular attributes such as kinetic chains(kc), branches (br), and rungs (rg) are expressed below using massaction rate laws derived from the kinetics scheme. A shorthand notationis used to define the concentration of catalyst with un-bridgedmolecules (ξ_(0,0)) and the concentration of catalyst with bridgedpolymer molecules (μ₀). Therefore the total catalyst concentration isξ_(0,0)+μ₀.

$\begin{matrix}{R_{kc} = {{{2\Omega{\sum\limits_{n = 1}^{\infty}{\sum\limits_{m = 1}^{\infty}P_{n,m}}}} + {2\Omega{\sum\limits_{n = 1}^{\infty}S_{n}}}} = {2{\Omega\left( {\xi_{0,0} + \mu_{0}} \right)}}}} & (11) \\{R_{br} = {{4\Psi{\sum\limits_{n = 1}^{\infty}{\sum\limits_{m = 1}^{\infty}P_{n,m}}}} = {4\Psi\xi_{0,0}}}} & (12) \\{{R_{rg} = {4\Psi{\sum\limits_{n = 1}^{\infty}{\sum\limits_{n = 1}^{\infty}{Pn}}}}},{{m + {4\Psi{\sum\limits_{n = 1}^{\infty}{Sn}}}} = {4{\Psi\left( {\xi_{0,0} + \mu_{0}} \right)}}}} & (13) \\{\xi_{0,0} = {{\sum\limits_{n = 1}^{\infty}{\sum\limits_{m = 1}^{\infty}P_{n,m}}} = {{\sum\limits_{n = 1}^{\infty}R_{n}} = {\sum\limits_{n = 1}^{\infty}L_{n}}}}} & (14) \\{\mu_{0} = {\sum\limits_{n = 1}^{\infty}{Sn}}} & (15)\end{matrix}$

The first step in rendering a usable model is to implement the“steady-state assumption” on the distributions of the propagatingpolymer species by setting the relevant polymer subspecies rates (R_(P)_(n,m) , R_(S) _(n) , R_(L) _(n) , R_(R) _(n) ) to zero. This is a verycommon assumption in addition polymerization modeling when the lifetimeof propagating chain is a very small fraction of the time period ofinterest. In most non-living commercial polymerizations of this type,the chain lifetime is typically much less than a second while a reactorresidence time is at least several minutes. The following relation isderived after implementing the “steady-state” assumption and summing thelive rates over all indices.

$\begin{matrix}{{{2\Psi\mspace{11mu}\xi_{0,0}} = {\Omega\mu}_{0}},{{{therefore}\mspace{14mu}\xi_{0,0}} = {\frac{\Omega}{\Omega + {2\Psi}}\left( {\xi_{0,0} + \mu_{0}} \right)}}} & (16)\end{matrix}$

The “steady-state assumption” results in relations for simple branchingmetrics (B_(c), B_(n), R_(c)) that will be useful in the moleculararchitecture model. In this particular case, instantaneous propertiesare convenient and relevant because they apply to a variety of reactorssuch as a steady state, well-mixed reactor or a batch reactor withnegligible drift in temperature or composition. The instantaneousbranching metrics (B_(c), B_(n), R_(c)) are equivalent to theircumulative average values when there is no spatial or temporal variationin chain transfer (Ω) and diene bridging rate (Ψ) parameters.

$\begin{matrix}{{{Instantaneous}\mspace{14mu}{Tetra}\text{-}{Functional}\mspace{14mu}{Branches}\mspace{14mu}{per}\mspace{14mu}{Kinetic}\mspace{14mu}{Chain}},{B_{c} = {\frac{R_{br}}{R_{kc}} = \frac{2\Psi}{\Omega + {2\Psi}}}}} & (17) \\{{{Instantaneous}\mspace{14mu}{Tetra}\text{-}{Functional}\mspace{14mu}{Branches}\mspace{14mu}{per}\mspace{14mu}{Polymer}\mspace{14mu}{Molecule}},{B_{n} = {\frac{R_{br}}{R_{\lambda_{0}}} = \frac{2\Psi}{\Omega}}}} & (18) \\{{{Instantaneous}\mspace{14mu}{Rungs}\mspace{14mu}{per}\mspace{14mu}{Kinectic}\mspace{14mu}{Chain}},{R_{c} = {\frac{R_{rg}}{R_{kc}} = \frac{2\Psi}{\Omega}}}} & (19)\end{matrix}$

Moments for Prediction of MWD Averages

A model describing the moments of the polymer species chain lengthdistributions can often be derived from population balances resultingfrom a kinetics scheme. A moment based model is useful in predictingmolecular weight averages and polydispersity index but in general doesnot describe smaller nuances in MWD such as bimodality, peak MW, andtailing. The method of moments entails the definition of variouspolymeric subspecies chain length distribution moments such as thosebelow. The bulk polymer moments (λ_(i)) reflect bulk polymer propertiesand solution of a model of bulk moments generally requires solution ofvarious living polymer moments.

$\begin{matrix}{{Living}\mspace{14mu}{Polymer}\mspace{14mu}{MWD}\mspace{14mu}{Moments}\text{:}\begin{matrix}{\mu_{i} = {\sum\limits_{n = 1}^{\infty}{n^{i}S_{n}}}} & {\xi_{i,j} = {\sum\limits_{n = 1}^{\infty}{\sum\limits_{m = 1}^{\infty}{n^{i}m^{j}P_{n,m}}}}}\end{matrix}} & (20) \\{{{{Bulk}\mspace{14mu}{Polymer}\mspace{14mu}{MWD}\mspace{14mu}{Moments}\text{:}}\quad}{\quad{\lambda_{i} = {{\sum\limits_{n = 1}^{\infty}{n^{i}\left( {D_{n} + S_{n} + L_{n} + R_{n}} \right)}} \approx {\sum\limits_{n = 1}^{\infty}{n^{i}D_{n}}}}}}} & (21)\end{matrix}$

Any skilled polymer reaction engineer would understand the derivationsof a Moments Model (Equations (20) and (21)) from a series of populationbalances. Rates of change of the leading bulk polymer moments (λ₀, λ₁,λ₂) are given below with negligible terms removed after imposing theassumption that kinetic chains are long, and therefore Φ>>Ω.

$\begin{matrix}{{{R_{\lambda_{0}} = {{2{\Omega\left( {\mu_{0} + \xi_{0,0}} \right)}} - {4\Psi\xi_{0,0}}}}\mspace{11mu}\quad}{\quad{R_{\lambda_{1}} = {{2{\Phi\left( {\mu_{0} + \xi_{0,0}} \right)}\mspace{11mu} R_{\lambda_{2}}} = {{2{\Phi\left( {\xi_{1,0} + \xi_{0,1} + {2\mu_{1}}} \right)}} + {8\Psi\xi_{1,1}}}}}}} & (22)\end{matrix}$

Evaluation of the rates of change of these bulk moments requires anumber of living polymer subspecies moments. These live polymer momentsare algebraic quantities because of the “steady-state assumption” andare given below. Additional live moments are required when higher bulkmoments such as λ₃ are predicted.

$\begin{matrix}{\xi_{0,0} = {{\frac{\Omega\mu_{0}}{2\Psi}\mspace{11mu}\xi_{1,0}} = {\xi_{0,1} = {{\frac{{\Phi\xi_{00}} + {\Omega\mu}_{1}}{\Omega + {4\Psi}}\mspace{11mu}\xi_{1,1}} = {{\frac{\Phi\left( {\xi_{1,0} + \xi_{0,1}} \right)}{{2\Omega} + {4\Psi}}\mspace{11mu}\mu_{1}} = \frac{{\Phi\mu}_{0} + {2{\Psi\left( {\xi_{1,0} + \xi_{0,1}} \right)}}}{\Omega}}}}}} & (23)\end{matrix}$

The instantaneous number and weight average chain lengths (DPn, DPw) areprovided below, after algebraic simplification of the moment rates. Ofcourse, the average molecular weights (M_(n), M_(w)) are equal to theaverage chain lengths multiplied by the apparent monomeric repeat unitweight in g/mole.

$\begin{matrix}{{DP_{n}} = {\frac{R_{\lambda_{1}}}{R_{\lambda_{0}}} = {{\frac{\Phi\left( {{2\Psi} + \Omega} \right)}{\Omega^{2}}\mspace{11mu} Z_{p}} = {\frac{DP_{w}}{DP_{n}} = {\frac{R_{\lambda_{2}}R_{\lambda_{0}}}{\left( R_{\lambda_{1}} \right)^{2}} = \frac{2\left( {{8\Psi^{2}} + {10\Psi\Omega} + \Omega^{2}} \right)}{\left( {{2\Psi} + \Omega} \right)^{2}}}}}}} & (24)\end{matrix}$

The expression of the model is further simplified by a fewsubstitutions, such as the average linear kinetic chain length DP_(no),being equal to Φ/Ω. Also, the model can be expressed as a function ofany of the instantaneous branching metrics B_(c), B_(n), and R_(c). Themodel is written below in terms of diene “Ladder Branches” per kineticchain (B_(c)) and branches per polymer molecule (B_(n)). It waspreviously shown that branches per polymer molecule equals rungs perkinetic chain (B_(n)=R_(c)) for this system.

$\begin{matrix}{{DP_{n}} = {\frac{DP_{n\; o}}{1 - B_{c}} = {D{P_{n\; o}\left( {1 + B_{n}} \right)}}}} & (25) \\{Z_{p} = {\frac{DP_{w}}{DP_{n}} = {\frac{M_{w}}{M_{n}} = {{2 + {6B_{c}} - {4B_{c}^{2}}} = \frac{2 + {10B_{n}} + {4B_{n}^{2}}}{\left( {1 + B_{n}} \right)^{2}}}}}} & (26)\end{matrix}$

The number and weight average molecular weights (M_(n), M_(w)) can alsobe predicted as functions of diene “Ladder Branches” per kinetic chain(B_(c)) or branches per polymer molecule (B_(n)) after defining thenumber and weight average linear kinetic chain weight as M_(no)) andM_(wo).

$\begin{matrix}{\mspace{76mu}{M_{n} = {\frac{M_{no}}{1 - B_{c}} = {M_{n\; o}\left( {1 + B_{n}} \right)}}}} & (27) \\{M_{w} = {{M_{wo}\frac{2 + {6B_{c}} - {4B_{c}^{2}}}{1 - B_{c}}} = {{M_{wo}\frac{2 + {10B_{n}} + {4B_{n}^{2}}}{1 + B_{n}}\mspace{14mu}{where}\mspace{14mu} M_{wo}} = {2\mspace{11mu} M_{no}}}}} & (28)\end{matrix}$

An unexpected prediction arising from the Moments Model (Equations (20)and (21)) is that at high dienes branching levels, the maximumpolydispersity is about 4. Of course, this prediction is for an idealco-polymerization and a single symmetric catalyst system and anynon-idealities are likely to give an increased polydispersity.

Model of the Complete MWD Curve

At times, it is possible to solve population balances for a molecularweight distribution curve. Explicit algebraic solutions are normallyonly available for instances of no spatial or temporal variations inreaction rates, such as assumed in this case. The solution begins withthe definition of yet another distributional quantity Vn derived fromPn,m. The population balance for Vn is derived by summing over thepopulation balance for Pn,m, with simplification due to symmetry.

$\begin{matrix}{V_{\mathfrak{n}} = {{\sum\limits_{m = 1}^{n - 1}P_{m,{n - m}}} = {\sum\limits_{m = 1}^{n - 1}P_{{n - m},m}}}} & (29) \\{R_{V_{n}} = {0 = {{2\Phi\mspace{11mu}\left( {V_{n - 1} - V_{n}} \right)} - {\left( {{2\Omega} + {4\Psi}} \right)V_{n}} + {2{\Omega\left( {S_{n} + L_{n}} \right)}}}}} & (30)\end{matrix}$

Due to the assumption of long chains, it is possible to treat allsubspecies distributions as if they were continuous rather than discretefunctions. The discrete steady-state polymer species population balancescan be closely approximated by differential equations in the continuousvariable n when difference terms are replaced by derivatives. Forexample, the steady-state population balance for Sn contains thedifference term Sn-Sn−1 which is replaced by the derivative as shown inequation (31).

$\begin{matrix}{{\Phi\left( {S_{n} - S_{n - 1}} \right)} \sim {\Phi\frac{d{S(n)}}{dn}}} & (31)\end{matrix}$

Similar replacements result in the following series of ordinarydifferential equations (ODEs) which can be integrated to yield the chainlength distributions of the various defined live subspeciesdistributions L(n), S(n), and V(n). The model is summarized below as aninitial value problem, where the chain length distribution functions areassumed to start at n=0. The lower limit of n=0 for the distributionfunctions is chosen for mathematical simplicity alone and ultimatelymakes no significant impact on model predictions when high polymers areformed.

$\begin{matrix}{{\Phi\frac{{dL}(n)}{dn}} = {{{{- \left( {\Omega + {4\Psi}} \right)}\mspace{11mu}{L(n)}} + {\Omega\mspace{11mu}{S(n)}\mspace{11mu}{L(0)}}} = {{\xi_{0,0}\left( {\Omega + {2\Psi}} \right)}/\Phi}}} & (32) \\{{2\Phi\frac{{dS}(n)}{dn}} = {{{{- 2}\Omega\mspace{11mu}{S(n)}} + {4\Psi\mspace{11mu}{V(n)}{\mspace{11mu}\;}{S(0)}}} = 0}} & (32) \\{{2\Phi\frac{{dV}(n)}{dn}} = {{{{- \left( {{2\Omega} + {4\Psi}} \right)}\mspace{11mu}{V(n)}} + {2\Omega\mspace{11mu}{S(n)}} + {2\Omega\mspace{11mu}{L(n)}\mspace{11mu}{V(0)}}} = 0}} & (33)\end{matrix}$

The instantaneous dead polymer chain length distribution is proportionalto L_(n), as evident from the species rate (R_(D) _(n) ). Thereforethrough L_(n) the solution of the above system of differential equationsgives the instantaneous dead polymer distribution X_(n) and thecontinuous distribution X(n) is similarly proportional to L(n).

$\begin{matrix}{{{Instantaneous}\mspace{14mu}{Dead}\mspace{14mu}{Polymer}\mspace{14mu}{Distribution}},{X_{n} = {{R_{D_{n}}/{\sum\limits_{m = 1}^{\infty}R_{D_{m}}}} = {{L_{n}/{\sum\limits_{m = 1}^{\infty}L_{m}}} = {L_{n}/\xi_{0,0}}}}}} & (34)\end{matrix}$

Solution for the Complete MWD Curve

The distribution functions of the increasing polymer chain length can besolved either numerically or analytically by persons familiar with theintegration of ordinary differential equations. The analytical solution,although algebraically complicated, is given here because it agreescompletely with the Moments Model (Equations (20) and (21)) while alsopredicting nuances in MWD such as peak location multi-modality, andtailing.

The software package known as Mathematica™ was used to develop ananalytical solution to the system of ordinary differential equationsthat describe the propagating polymer distribution functions L(n), S(n),and V(n). The analytical solution for L(n) was used to describe theinstantaneous dead polymer distribution X(n), by normalizing L(n) overits integral.

$\begin{matrix}{{{X(n)} = {{L(n)}/{\int\limits_{0}^{\infty}{L(m)}}}}\mspace{11mu}{dm}} & (35)\end{matrix}$

An explicit analytical solution for X(n) may be obtained usingMathematica™. The analytical solution for X(n) is described below as afunction of parameters B_(n) and DP_(no), and the solution may berestated in terms of R_(c) or B_(n) through the substitutionR_(c)=B_(n)=B_(c)/(1−B_(c)). (36)

The chain length distribution function X(n) is evaluated as follows fromthe definition of RootSum given by Mathematica™. The polynomial belowhas three roots which will be called x₁, x₂, and x₃. Two of the threeroots of the polynomial are complex over the range of possible values ofB_(n).

0=1+B _(n)+(3+5B _(n)+2B _(n) ²)x+3(1+B _(n))x ² +x ³  (37)

The roots x₁, x₂, and x₃ are used in the instantaneous dead chain lengthdistribution function X(n).

$\begin{matrix}{{X(n)} = {\left( \frac{1 + B_{n}}{DP_{no}} \right){\sum\limits_{i = 1}^{3}{\frac{\left( {1 + {2x_{i}} + {B_{n}x_{i}} + x_{i}^{2}} \right)}{3 + {5B_{n}} + {2B_{n}^{2}} + {6\left( {1 + B_{n}} \right)x_{i}} + {3x_{i}^{2}}}e^{x_{i}{n/D}P_{no}}\mspace{11mu}\left( {0 \leq n \leq \infty} \right)}}}} & (38)\end{matrix}$

Various moments of X(n) are evaluated to give instantaneous number andweight average chain lengths (DP_(n), DP_(w)) or molecular weights(M_(n), M_(w)). The average chain lengths and weights resulting from thecontinuous distribution X(n) are equal to the moment model predictionsgiven previously for long chain polymerization and a discretedistribution and are expressed below in terms of both B_(c) and B_(n),where R_(c)=B_(n).

$\begin{matrix}{{DP}_{n} = {\frac{\overset{\infty}{\int\limits_{0}}{{{nX}(n)}{dn}}}{\overset{\infty}{\int\limits_{0}}{{S(n)}\mspace{11mu}{dn}}} = {\frac{{DP}_{no}}{1 - B_{c}} = {{{{DP}_{no}\left( {1 + B_{n}} \right)}\mspace{11mu} M_{n}} = {\frac{M_{no}}{1 - B_{c}} = {M_{no}\left( {1 + B_{n}} \right)}}}}}} & (39) \\{Z_{p} = {\frac{{DP}_{w}}{{DP}_{n}} = {\frac{M_{w}}{M_{n}} = {\frac{\left( {\overset{\infty}{\int\limits_{0}}{n^{2}{X(n)}\mspace{11mu}{dn}}} \right)\left( {\overset{\infty}{\int\limits_{0}}{{X(n)}\mspace{11mu}{dn}}} \right)}{\left( {\overset{\infty}{\int\limits_{0}}{{{nX}(n)}\mspace{11mu}{dn}}} \right)} = {{2 + {6\mspace{11mu} B_{c}} - {4\mspace{11mu} B_{c}^{2}}} = \frac{2 + {10\mspace{11mu} B_{n}} + {4\mspace{11mu} B_{n}^{2}}}{\left( {1 + B_{n}} \right)^{2}}}}}}} & (40)\end{matrix}$

Those skilled in the art of polymer reaction engineering are familiarwith the use of predicted bulk polymer MWD models to create simulatedSize Exclusion Chromatography (SEC) curves. Such a simulation is usefulin relating how kinetics and recipes are expected to impact SECmeasurements. The primary calibrated result of an SEC measurement is atable or plot of dw/d Log(M) versus Log(M), where M is species molecularweight or size and dw/d Log(M) is an indication of the relative amountof polymer corresponding to M. It is commonly accepted that this SECresult can be simulated by a table or plot of n² X(n) versus Log(M),where n² X(n) is expected to be proportional to dw/d Log(M).

FIG. 2 shows a series of simulated SEC curves wherein the level of diene“Ladder Branching” (B_(c), B_(n), R_(c)) is varied. The independentvariable in FIG. 2 is scaled by linear molecular weight or chain lengthsuch that the plotting is universal and independent of startingmolecular weight. The zero-branching case in FIG. 2 is the well-known“most probable” MWD (P. J. Flory, J. Am. Chem. Soc. 1936, 58, 1877) andis the expected MWD for linear addition co-polymerization performedunder ideal homogeneous conditions.

A more detailed analysis of the peak MW values has been performed usinga large series of branching levels applied to the MWD model. FIG. 3shows a universal plot of relative peak MW as a function branchinglevel. FIG. 3 demonstrates the low branching region of peak MWinsensitivity (0<R_(c)<0.15) as well as the higher branching regime(R_(c)≥0.15) wherein the peak MW increases steadily with branchinglevel.

Alternate Tri-Functional Dienes “Ladder Branching” Mechanism and Model

There are alternate mechanisms that can explain branching and MWD trendsobserved when dual chain catalysts incorporate dienes under desiredconditions. While M_(n) is often observed to increase with dienesaddition, some catalyst-diene combinations have been found to result inM_(w) increases while demonstrating little or no measurable M_(n)increase as diene levels are raised. One explanation for constant M_(n)is that a single beta-hydride elimination (or chain-transfer tohydrogen) might tend to occur immediately after the diene has insertedon both propagating chains. This scenario would result in the creationof tri-functional branches by the dienes insertion and, in pure form,would eliminate bridged propagating species (S_(n)) from the kinetics.

The kinetics scheme is modified to consider this alternative mechanismby substituting the following reactions for “Dienes Bridging”.

Reaction Rate Constant Tri-functional P_(n,m) + D → P_(0,n+m) + br + kc2 k_(d), (L/mole/sec) Diene Branching P_(n,m) + D → P_(n+m,0) + br + kc2 k_(d), (L/mole/sec)

Any polymer reaction engineer skilled in the art of modeling andkinetics could re-derive the moments and MWD function model for thesealternative kinetics using the same sequence of assumptions as before.The resulting instantaneous dead chain length distribution function X(n)is given below for this tri-functional branching mechanism

$\begin{matrix}{{X(n)} = {\left( \frac{\begin{matrix}{{{Cosh}\left\lbrack \frac{n\sqrt{B_{c}}}{{DP}_{no}\left( {1 - B_{c}} \right)} \right\rbrack} -} \\{\sqrt{B_{c}}{{Sinh}\left\lbrack \frac{n\sqrt{B_{c}}}{{DP}_{no}\left( {1 - B_{c}} \right)} \right\rbrack}}\end{matrix}}{{DP}_{no}\left( {1 - B_{c}} \right)} \right){e^{{- n}/{({{DP}_{no}{({1 - B_{c}})}})}}\;\left( {0 \leq n \leq \infty} \right)}}} & (41)\end{matrix}$

In Equation (41), B_(c) is defined as branch points per kinetic chainand DP_(no) is defined as the diene-free average linear chain length.The kinetic scheme assumes that the linear (kinetic) chain lengthactually decreases with dienes incorporation due to diene induced betahydride elimination. Therefore, a good alternate indication of branchingis B_(n), which is defined as branch points per number average polymermolecule, where B_(c)=B_(n)/(1+B_(n)). The function X(n) is easilyrewritten in terms of B_(n).

Integrations of X(n) gives results for instantaneous number and weightaverage chain lengths (DP_(n), DP_(w)) or molecular weights (M_(n),M_(w)). The average chain lengths and weights resulting from thecontinuous distribution X(n) are equal to a moment model predictionswhen long chain polymerization is assumed. Integration of X(n) confirmsthat DP_(n) and M_(n) are constant with respect to branching level(B_(c) or B_(n)). Integration of X(n) also shows how polydispersity isexpected to vary with branching level when dienes are assumed to createtri-functional branches.

$\begin{matrix}{{DP_{n}} = {\frac{\int\limits_{0}^{\infty}{n{X(n)}dn}}{\underset{0}{\int\limits^{\infty}}{{X(n)}dn}} = {{DP_{no}\mspace{14mu}{therefore}\text{:}\mspace{14mu} M_{n}} = M_{no}}}} & (42) \\{Z_{p} = {\frac{DP_{w}}{DP_{n}} = {\frac{M_{w}}{M_{n}} = {\frac{\begin{matrix}\left( {\int\limits_{0}^{\infty}{n^{2}{X(n)}dn}} \right) \\\left( {\int\limits_{0}^{\infty}{{X(n)}dn}} \right)\end{matrix}}{\left( {\int\limits_{0}^{\infty}{n{X(n)}dn}} \right)^{2}} = {{2\left( {1 + B_{c}} \right)} = \frac{\begin{matrix}2 \\\left( {1 + {2B_{n}}} \right)\end{matrix}}{1 + B_{n}}}}}}} & (43)\end{matrix}$

The above relationship of polydispersity (M_(w)/M_(n)) to tri-functionalbranching level shows no instability or divergence at any branchinglevel. Most surprising is that at high branching levels thepolydispersity is predicted to level off at 4. Of course, thisprediction is for an ideal co-polymerization and symmetric catalystsystem with any non-idealities expected to give an increasedpolydispersity.

The chain length distribution function can again be used to constructpredicted MWD curves. FIG. 4 is a series of simulated SEC curves whereinthe level of tri-functional branching (B_(c) or B_(n)) is varied. Theindependent variable in FIG. 4 is scaled by linear molecular weight orchain length such that the plotting is universal and independent ofstarting molecular weight. The zero-branching case in FIG. 4 is thewell-known “most probable” MWD and is expected for linear additionco-polymerization performed under ideal homogeneous conditions. FIG. 5is a plot of relative peak MW for tri-functional dienes branching whichdemonstrates that the MWD peak is most sensitive to branching level atintermediate branching levels in the approximate range of 0.2<B_(n)<0.9or 0.17<B_(c)<0.5.

Conventional Branching Models

The purpose of this section is to compare a variety of conventionaldienes branching and random polymer coupling to the “Ladder Branching”models. The comparison demonstrates the inherent instability ofconventional dienes branching and random polymer coupling in contrast to“Ladder Branching”. The molecular architecture resulting from the dienes“Ladder Branching” is different from (a) the conventional DienesContinuous Stirred Tank Reactor (CSTR) Branching Model, (b) conventionalDienes Semi-Batch Branching Model; (c) Polymer CSTR Coupling Model; and(d) Polymer Batch Coupling Model.

a) Conventional Dienes CSTR Branching Model Ver Strate-1980 (G. VerStrate, C. Cozewith, W. W. Graessley, J. App. Polym. Sci. 1980, 25, 59),Guzman-2010 (J. D. Guzman, D. J. Arriola, T. Karjala, J. Gaubert, B. W.S. Kolthammer, AIChE 2010, 56, 1325):

$\begin{matrix}{Z_{p} = {{\frac{DP_{w}}{DP_{n}}\frac{M_{w}}{M_{n}}} = {{\left( {1 - B_{c}} \right)\left( \frac{\begin{matrix}{1 - {4\; B_{c}} -} \\\sqrt{1 - {8\; B_{c}}}\end{matrix}}{\left( {2B_{c}} \right)^{2}} \right)} = \frac{\begin{matrix}{1 - {3B_{n}} -} \\\sqrt{\begin{matrix}\left( {1 - {7B_{n}}} \right) \\\left( {1 + B_{n}} \right)\end{matrix}}\end{matrix}}{\left( {2B_{n}} \right)^{2}}}}} & (44)\end{matrix}$

b) Conventional Dienes Semi-Batch Branching Model, Cozewith-1979 (C.Cozewith, W. W. Graessley, G. Ver Strate, Chem. Eng. Sci. 1979, 34,245), and d) Polymer Batch Coupling Model, Cozewith-1979, Flory-1953 (P.J. Flory, Principles of Polymer Chemistry, Cornell University Press,1953), Tobita-1995 (H. Tobita, J. Polym. Sci. B 1995, 33, 1191):

$\begin{matrix}{Z_{p} = {\frac{DP_{w}}{DP_{n}} = {\frac{M_{w}}{M_{n}} = {\frac{2 - {2B_{c}}}{1 - {4B_{c}}} = \frac{2}{1 - {3B_{n}}}}}}} & (45) \\{{X(n)} = {{e^{{{- {n{({1 + {2B_{c}}})}}}/D}P_{no}}\left( \frac{1}{DP_{no}} \right)}{\sum\limits_{b = 0}^{\infty}{\left( \frac{n}{DP_{no}} \right)^{3b}\frac{\left( {2B_{c}} \right)^{b}}{\begin{matrix}{\left( {1 + b} \right)!} \\{\left( {1 + {2b}} \right)!}\end{matrix}}}}}} & (46)\end{matrix}$

c) Polymer CSTR Coupling Model:

$\begin{matrix}{Z_{p} = {\frac{DP_{w}}{DP_{n}} = {\frac{M_{w}}{M_{n}} = {{\left( {1 - B_{c}} \right)\left( \frac{\begin{matrix}{1 -} \\\sqrt{\begin{matrix}{1 -} \\{16\; B_{c}}\end{matrix}}\end{matrix}}{4\; B_{c}} \right)} = \frac{1 - \sqrt{\begin{matrix}{\left( {1 - {15B_{n}}} \right)\text{/}} \\\left( {1 + B_{n}} \right)\end{matrix}}}{4B_{n}}}}}} & (47)\end{matrix}$

Characterizing Tetra-Functional Long-Chain Branched Polyolefin

Depending on the degree of branching, a variety of methods can eitherdetermine LCB, such as nuclear magnetic resonance (NMR), or distinguishthe effect of LCB in the polymer. For example, the effect of LCB isobserved in shear flow in the van Gurp-Palmen analysis, also an increaseof the shear viscosity at low angular frequencies and strength of theshear thinning behavior can be attributed to LCB. In extensional flow,the influence of LCB is usually identified in the degree of hardening orthe strength of the melt and the maximum deformation achieved. Otherplots such as Mark-Houwink plots, broadening molecular weightdistributions (MWD), and g′_(vis) plots provide additional informationabout LCB. A high level of natural LCB in a polymer is difficult toachieve due to the limited concentration of vinyl terminated polymers(maximum one per polymer chain) and the need to run to high conversionto ensure LCB formation. To ensure high conversion, there is a lowethylene concentration in the reactor, thus enabling a great amount ofvinyl terminated polymers to be reinserted in a second polymer chain.

The conventional process of incorporating dienes into a polymersynthesis system suffers from the fundamental flaw of gel formation orreactor fouling at high branching levels. The kinetic modeling,discussed in previous paragraphs, may provide good predictive resultsthat enable a better understanding of gel formation. For example, longerpolymer chains have proportionally more pendant vinyls and polymerchains containing more pendant vinyls will more likely re-insert intothe catalyst to form a LCB. Thus, the larger polymer chainspreferentially re-insert forming tetra-functional branches, which areeven larger polymer molecules, and a gel problem or instability resultswhen the LCB level reaches a threshold value. A simulation of the weightaverage molecular weight (M_(w)) and number average molecular weight(M_(n)) as a function of conventional tetra-functional branching isshown in FIG. 1 for ethylene-based polymer in a semi-batch reactor atconstant pressure. In FIG. 1, M_(n) only marginally increases as M_(w)becomes infinite. In this example, as the M_(w) increases to a numbergreater than 200,000 grams per mole (g/mol), the polymer molecularweight distribution (MWD) becomes unstable and gels begin to form. TheMWD is defined by the weight average molecular weight, M_(w), divided bythe number average molecular weight, M_(n), (M_(w)/M_(n)).

Polymer gels are narrowly defined for the purpose of this disclosure tobe a polymer fraction that is phase separated due to its high branchinglevel and/or high molecular weight. Polymer gels can be observed insolution or in the melt and tend to interfere with properties such asoptical clarity and film and fiber performance. Polyethyleneinterpolymer gels can be measured by degree of polymer insolubility inhot xylene. Gels content is often correlated to and therefore estimatedfrom GPC polymer recovery percentage. When polymer gels form, they maydeposit within the reactor and result in fouling.

FIG. 7 and FIG. 8 show the differences in the MWD curves expected fromconventionally branched and “Ladder Branched” polymers. A series ofmetrics describing MWD characteristics has been developed from a studyof MWD data and comparison to MWD models. Each of the MWD descriptivemetrics presented here is independent of average MW and is focused onthe high MW portion of the MWD. The MWD metrics are derived from ascaled MWD curve (dW/dlogM) with the primary or highest peak of the MWDdefined as having a value of unity. If more than one peak have the sameheight, the highest MW peak is the primary peak. The independentvariable in the MWD curve is Log(M), which is the logarithm of M to base10. The metrics will be defined and presented as a function ofM_(w)/M_(wo) and M_(p)/M_(po) which can be translated to branches permolecule or segment using FIG. 6, FIG. 7, FIG. 8, and FIG. 9. Oneskilled in the art of GPC data interpretation would understand thesemetrics and would be able to calculate them from GPC data.

A family of GPC shape metrics, G(A/B), are calculated from the slopes atdefined points on the right hand side of the MWD curve, where S(A) andS(B) are the first occurrences of these slopes to the right of theprimary peak at A % and B % of the height of the primary peak. Thepoints A and B are selected as pairs that would have nearly the sameslope if the MWD were “most probable”. A depiction of these points andtheir slope is shown in the graph of FIG. 10 for a most probable MWD.These slope pairs S(A) and S(B) are used together to calculate afunction G(A/B) resembling a second derivative, which will be shown tobe a useful metric to differentiate “Ladder Branched” MWDs fromconventionally or randomly branched MWDs. Values of G(79/29) andG(96/08) describe the change in slope of the right hand side (RHS) ofthe MWD and are defined below from the high MW slopes:

G(79/29)=(S(79)−S(29))/S(79)  (48)

G(96/08)=(S(96)−S(8))/S(96)  (49)

The shape metrics G(79/29) and G(96/08) are tested on the MWD models fortetra-functional “Ladder Branching” and conventional dienes branchingwith the results plotted in FIG. 11, FIG. 12, FIG. 13, FIG. 14. Thefigures indicate that conventional branching gives G(79/29) and G(96/08)values that increase steadily as MW responds to branching. However, whenapplied to “Ladder Branching”, these shape metrics fall precipitously atlow levels of branching (low M_(w)/M_(po)) then approach zero atmoderate to high levels of branching. That is not surprising, since thehigh MW portion of a “Ladder Branched” MWD resembles a most probableMWD.

FIG. 11, FIG. 12, FIG. 13, and FIG. 14 depict a similar response ofG(79/29) and G(96/08) metrics to branching, however, the G(96/08) metricis expected to be more sensitive to high MW tailing that would resultfrom conventional dienes branching. The term “high MW tailing” or “highmolecular weight tail” refers to the high molecular weight fraction asshown by the conventional GPC and the absolute GPC. Depending oncatalyst-diene pairing and experimental conditions, one might expect a“Ladder Branched” system to have some conventional branching therebyraising the shape metric value above that expected for pure “LadderBranching”.

MWD Area Metrics

Visual inspections of the “Ladder Branched” MWD show that there is acharacteristic lack of a high MW tail normally seen for branchedpolymers. FIG. 16 and FIG. 17 demonstrate how the model predicts a lackof tailing for “Ladder Branched” polymers. The “Ladder Branching” MWDdata shows the characteristic lack of tail for many experiments but alsoindicates some tail formation is possible depending on polymerizationconditions and diene/catalyst pairing.

Polydispersity indices (M_(w)/M_(n), M_(z)/M_(w), etc) are known metricsfor tailing, but are not preferred due to their sensitivity to low MWDartifacts. Therefore, a more focused version of the polydispersityindices is used to develop a standard for which the integrals areperformed only on the high MW portion of the MWD. The M_(w)/M_(n) andM_(z)/M_(w) metrics are successful in differentiating dienes “LadderBranching” from conventional branching and are very sensitive to high MWbaseline selection and baseline noise.

The area under the MWD curve is relatively insensitive to baselineissues as compared to the higher moments required to calculate MWDdispersity indices (M_(w)/M_(n), M_(z)/M_(w), etc). Therefore, it wasdecided that metrics be developed which entail the non-weightedintegration of the MWD. These MWD area metrics, A_(HIGH) and A_(TAIL),are calculated from the GPC curve areas for defined regions on the righthand side of the MWD curve. The MWD area metrics (A_(HIGH) and A_(TAIL))are derived from a scaled MWD curve (dW/log M) with the primary orhighest peak of the MWD defined as having a value of unity. If more thanone peak have the same height, the highest MW peak is the primary peak.The independent variable in the MWD curve is Log(M), which is thelogarithm of M to base 10. Both of the MWD area metrics depend on thepoint of maximum slope of the high MW portion of the MWD. The quantitiesand limits necessary for evaluating the area metrics are listed below,and demonstrated in FIG. 15 for a most probable MWD.

S_(max)=first instance of a maximum downward slope on the RHS (higher MWside) of the primary peak (absolute value of the slope) of the scaledMWD

H_(smax)=height of the scaled MWD at the point of maximum slope

pt1=Log M value of S_(max)

pt2=Log M value where the S_(max) tangent crosses the x-axis

The MWD area metrics are defined below, where A_(HIGH) is merely thearea of the MWD region falling after the point of maximum slope. Thesecond area metric, A_(TAIL), is the small high MW area depicted in FIG.15, and is evaluated by subtracting a triangular area from A_(HIGH).

$\begin{matrix}{A_{HIGH} = {\overset{\infty}{\int\limits_{p\; t\; 1}}{{MWD}\; d\;{Log}\; M}}} & (50) \\{A_{TAIL} = {A_{HIGH} - {1\text{/}2\left( H_{s\max} \right)^{2}\text{/}S_{\max}}}} & (51)\end{matrix}$

The area metrics A_(HIGH) and A_(TAIL) have been tested on the MWDmodels for “Ladder Branching” and conventional dienes branching with theresults plotted in FIG. 16, FIG. 17, FIG. 18, and FIG. 19. The plotsshow that the high MW area, as defined by A_(HIGH) or A_(TAIL),increases dramatically as the conventional branching level is increased.However, the “Ladder Branching” model predicts that the high MW areametrics (A_(HIGH) or A_(TAIL)) are almost unaffected by “LadderBranching” level. The values of A_(HIGH) and A_(TAIL) for a mostprobable MWD are about 0.07 and 0.015, respectively. Example MWD datawill demonstrate that the dienes-free linear polymers tend to haveslightly higher values of A_(HIGH) and A_(TAIL) due to non-ideal aspectsof the polymerization. Example data also show a variety of highlybranched “Ladder Branched” polymers with essentially no high MW tailbeyond what is expected from a most probable MWD. The high MW areametrics also are diagnostic of slight levels of high MW tail formationthat “Ladder Branched” polymer can exhibit when accompanied by a degreeof conventional branching. The metric A_(TAIL) is less influenced bylinear MWD non-ideality than A_(HIGH). However, in theory A_(HIGH) andA_(TAIL) metrics are equally indicative of high MW tail formation.

Tetra-functional Long-Chain Branched Polyolefin

Polymers produced from the “Ladder Branching”, as described in Scheme 4,are included in this disclosure.

In embodiments, the ethylene-based polymers of this disclosure include amelt viscosity ratio or rheology ratio (V_(0.1)/V₁₀₀) at 190° C. of atleast 10, where V_(0.1) is the viscosity of the ethylene-based polymerat 190° C. at an angular frequency of 0.1 radians/second, and V₁₀₀ isthe viscosity of the ethylene-based polymer at 190° C. at an angularfrequency of 100 radians/second. In one or more embodiments, the meltviscosity ratio is at least 14, at least 20, at least 25, or at least30. In some embodiments, the melt viscosity ratio is greater than 50, atleast 60, or greater than 100. In some embodiments, the melt viscosityratio is of from 14 to 200.

The “rheology ratio” and “melt viscosity ratio” are defined byV_(0.1)/V₁₀₀ at 190° C., where V_(0.1) is the viscosity of theethylene-based polymer at 190° C. at an angular frequency of 0.1radians/second, and V₁₀₀ is the viscosity of the ethylene-based polymerat 190° C. at an angular frequency of 100 radians/second.

In one or more embodiments, the ethylene-based polymers of thisdisclosure have an Average g′ less than 0.86, where the Average g′ is anintrinsic viscosity ratio determined by gel permeation chromatographyusing a triple detector. In some embodiments, the ethylene-basedpolymers of this disclosure have an Average g′ from 0.64 to 0.86. Allindividual values and subranges encompassed by “from 0.64 to 0.86” aredisclosed herein as separate embodiments; for example, the Average g′ ofthe ethylene-based polymer may range from 0.64 to 0.75, from 0.68 to0.79, or from 0.65 to 0.83. In one or more embodiments, the Average g′is from 0.65 to 0.84, from 0.66 to 0.82, or from 0.66 to 0.80.

In some embodiments, the ethylene-based polymers have a G(79/29) valueof less than or equal to 0.035 as determined from a gel permeationchromatography curve having a peak height, a slope M79 at 79% of thepeak height, and a slope M29 at 29% of the peak height, wherein theG(79/29) value equals (M79−M29)/M79. All individual values and subrangesencompassed by “of less than or equal to 0.035” are disclosed herein asseparate embodiments; for example, “of less than or equal to 0.035”includes from greater than 0.0 to 0.035, from 0.010 to 0.034, andincludes negative values. In one or more embodiments, the ethylene-basedpolymer of this disclosure may have a G(79/29) value of less than orequal to 0.030 as determined from a gel permeation chromatography curve.

In one or more embodiments, the melt viscosity ratio of theethylene-based polymer of this disclosure may be greater than ten timesthe elasticity factor where the melt viscosity ratio (V_(0.1)/V₁₀₀) isdetermined by V_(0.1), the viscosity of the ethylene-based polymer at190° C. at an angular frequency of 0.1 radians/second, and V₁₀₀, theviscosity of the ethylene-based polymer at 190° C. at an angularfrequency of 100 radians/second, and the elasticity factor m is [((tan(δ_(0.1))−tan (δ₁₀₀))*1000)/(0.1−100))], wherein tan (δ_(0.1)) is thetangent of the phase angle at 0.1 radians/second, and tan (δ₁₀₀) is thetangent of the phase angle at 100 radians/second.

In one or more embodiments, the ethylene-based polymer may have anelasticity factor m at 190° C. that is less than or equal to 8seconds/radian, where m is [((tan (δ_(0.1))−tan(δ₁₀₀))*1000)/(0.1-100))]. In other embodiments, the ethylene-basedpolymer may have an elasticity factor m at 190° C. that is less than orequal to 4 seconds/radian.

In various embodiments, the melt strength of the ethylene-based polymerof this disclosure may be greater than 6 cN (Rheotens device, 190° C.,2.4 mm/s², 120 mm from the die exit to the center of the wheels,extrusion rate of 38.2 s⁻¹, capillary die of 30 mm length, 2 mm diameterand 1800 entrance angle). In some embodiments, the melt strength of theethylene-based polymer may be greater than 10 cN.

In embodiments, the ethylene-based polymer may have a molecular weighttail quantified by an MWD area metric A_(TAIL), and A_(TAIL) is lessthan or equal to 0.04. All individual values and subranges encompassedby “less than or equal to 0.04” are disclosed herein as separateembodiments. For example, in some embodiments, the A_(TAIL) of theethylene-based polymer of this disclosure is greater than 0 and lessthan or equal to 0.03 as determined by gel permeation chromatographyusing a triple detector.

In embodiments, the M_(w) of ethylene-based polymer may be less than orequal to 800,000 Daltons, as determined by gel permeation chromatographyusing a triple detector. In one or more embodiments, the M_(w) of theethylene-based polymer may be less than or equal to 400,000 Daltons.

In various embodiments, the ethylene-based polymer may have anM_(p)/M_(p0) greater than 1.20, where M_(p) is the peak molecular weightof the ethylene-based polymer as determined from conventional gelpermeation chromatography, and M_(p0) is the initial peak molecularweight of the ethylene-based polymer without polyene comonomer.

In embodiments, the ethylene-based polymer has a M_(w)/M_(w0) of greaterthan 1.20, in which M_(w) is the weight average molecular weight of theethylene-based polymer as determined from a GPC curve of theethylene-based polymer acquired by gel permeation chromatography. M_(w0)is the initial weight average molecular weight of a comparativeethylene-based polymer by gel permeation chromatography. The comparativeethylene-based polymer is a reaction product of polymerizing ethylenemonomer and all C₃ to C₁₄ comonomers present in the ethylene-basedpolymer, if any, without the at least one polyene comonomer, under thedefined polymerization reaction conditions.

Each M_(w0) and the M_(p0) is a metric of polymer resins without theaddition of diene into the reactor during polymerization, as previouslydiscussed. Each subsequent addition of diene produces a polymer resinfrom which the metric M_(w) or M_(p) may be determined. The amount ofdiene incorporated into the reactor is small in comparison to the otherreactants in the reactor. Therefore, the addition of diene does notaffect the total amount of comonomer, ethylene, and solvent in thereactor.

In various embodiments, the ethylene-based polymer has a gpcBR branchingindex of from 0.1 to 3.0. All individual values and subrangesencompassed by “from 0.10 to 3.00” are disclosed herein as separateembodiments; for example, the ethylene-based polymers, may include agpcBR branching index of from 0.10 to 2.00, from 0.10 to 1.00, from 0.15to 0.65, from 0.20 to 0.75, or 0.10 to 0.95.

The long-chain branching polymerization processes described in thepreceding paragraphs are utilized in the polymerization of olefins,primarily ethylene and propylene. In some embodiments, there is only asingle type of olefin or α-olefin in the polymerization scheme, creatingwhat is essentially a homopolymer with small amounts of incorporateddiene comonomer. However, additional α-olefins may be incorporated intothe polymerization procedure. The additional α-olefin co-monomerstypically have no more than 20 carbon atoms. For example, the α-olefinco-monomers may have 3 to 10 carbon atoms or 3 to 8 carbon atoms.Exemplary α-olefin co-monomers include, but are not limited to,propylene, 1-butene, 1-pentene, 1-hexene, 1-heptene, 1-octene, 1-nonene,1-decene, 4-methyl-1-pentene, and ethylidene norbornene. For example,the one or more α-olefin co-monomers may be selected from the groupconsisting of propylene, 1-butene, 1-hexene, and 1-octene; or in thealternative, from the group consisting of 1-hexene and 1-octene.

The long-chain branched polymer, for example homopolymers and/orinterpolymers (including copolymers) of ethylene and optionally one ormore co-monomers such as α-olefins, may comprise at least 50 percent byweight of units derived from ethylene. All individual values andsubranges encompassed by “from at least 50 weight percent” are disclosedherein as separate embodiments; for example, the ethylene-basedpolymers, homopolymers and/or interpolymers (including copolymers) ofethylene and optionally one or more co-monomers such as α-olefins maycomprise at least 60 percent by weight of units derived from ethylene;at least 70 percent by weight of units derived from ethylene; at least80 percent by weight of units derived from ethylene; or from 50 to 100percent by weight of units derived from ethylene; or from 80 to 100percent by weight of units derived from ethylene.

In some embodiments of the ethylene-based polymers, the ethylene-basedpolymer includes additional α-olefin. The amount of additional α-olefinin the ethylene-based polymer is less than or equal to 50 mole percent(mol %); other embodiments the amount of additional α-olefin includes atleast 0.01 mol % to 25 mol %; and in further embodiments the amount ofadditional α-olefin includes at least 0.1 mol % to 10 mol %. In someembodiments, the additional α-olefin is 1-octene.

In some embodiments, the long-chain branched polymers may comprise atleast 50 percent by moles of units derived from ethylene. All individualvalues and subranges from at least 90 mole percent are included hereinand disclosed herein as separate embodiments. For example, the ethylenebased polymers may comprise at least 93 percent by moles of unitsderived from ethylene; at least 96 percent by moles of units; at least97 percent by moles of units derived from ethylene; or in thealternative, from 90 to 100 percent by moles of units derived fromethylene; from 90 to 99.5 percent by moles of units derived fromethylene; or from 97 to 99.5 percent by moles of units derived fromethylene.

In some embodiments of the long-chain branched polymer, the amount ofadditional α-olefin is less than 50%; other embodiments include at least1 mole percent (mol %) to 20 mol %; and in further embodiments theamount of additional α-olefin includes at least 5 mol % to 10 mol %. Insome embodiments, the additional (α-olefin is 1-octene.

Any conventional polymerization processes may be employed to produce thelong-chain branched polymer. Such conventional polymerization processesinclude, but are not limited to, solution polymerization processes, gasphase polymerization processes, slurry phase polymerization processes,and combinations thereof using one or more conventional reactors such asloop reactors, isothermal reactors, fluidized bed gas phase reactors,stirred tank reactors, batch reactors in parallel, series, or anycombinations thereof, for example.

In one embodiment, the ethylene based polymer may be produced viasolution polymerization in a dual reactor system, for example a singleloop reactor system, wherein ethylene and optionally one or moreα-olefins are polymerized in the presence of the catalyst system, asdescribed herein, and optionally one or more co-catalysts. In anotherembodiment, the ethylene-based polymer may be produced via solutionpolymerization in a dual reactor system, for example a dual loop reactorsystem, wherein ethylene and optionally one or more α-olefins arepolymerized in the presence of the catalyst system in this disclosure,and as described herein, and optionally one or more other catalysts. Thecatalyst system, as described herein, can be used in the first reactor,or second reactor, optionally in combination with one or more othercatalysts. In one embodiment, the ethylene-based polymer may be producedvia solution polymerization in a dual reactor system, for example a dualloop reactor system, wherein ethylene and optionally one or moreα-olefins are polymerized in the presence of the catalyst system, asdescribed herein, in both reactors.

In another embodiment, the long-chain branched polymer may be producedvia solution polymerization in a single reactor system, for example asingle loop reactor system, in which ethylene and optionally one or moreα-olefins are polymerized in the presence of the catalyst system, asdescribed within this disclosure, and optionally one or moreco-catalysts, as described in the preceding paragraphs. In someembodiments, the long-chain branching polymerization process forproducing the long-chain branched polymer includes polymerizing ethyleneand at least one additional α-olefin in the presence of a catalystsystem.

The long-chain branched polymers may further comprise one or moreadditives. Such additives include, but are not limited to, antistaticagents, color enhancers, dyes, lubricants, pigments, primaryantioxidants, secondary antioxidants, processing aids, UV stabilizers,and combinations thereof. The ethylene-based polymers may contain anyamounts of additives. The ethylene-based polymers may compromise fromabout 0 to about 10 percent by the combined weight of such additives,based on the weight of the ethylene based polymers and the one or moreadditives. The ethylene-based polymers may further comprise fillers,which may include, but are not limited to, organic or inorganic fillers.The long-chain branched polymers may contain from about 0 to about 20weight percent fillers such as, for example, calcium carbonate, talc, orMg(OH)₂, based on the combined weight of the ethylene based polymers andall additives or fillers. The ethylene-based polymers may further beblended with one or more polymers to form a blend.

In some embodiments, the long-chain polymerization process for producinglong-chain branched polymers may include polymerizing ethylene and atleast one additional α-olefin in the presence of a catalyst having twopolymer producing sites. The long-chain branched polymer resulting fromsuch the catalyst having two polymer producing sites may have a densityaccording to ASTM D792 (incorporated herein by reference in itsentirety) from 0.850 g/cm³ to 0.960 g/cm³, from 0.880 g/cm³ to 0.920g/cm³, from 0.880 g/cm³ to 0.910 g/cm³, or from 0.880 g/cm³ to 0.900g/cm³, for example.

In another embodiment, the long-chain branched polymer resulting fromthe long-chain polymerization process may have a melt flow ratio(I₁₀/I₂) from 5 to 100, in which melt index I₂ is measured according toASTM D1238 (incorporated herein by reference in its entirety) at 190° C.and 2.16 kg load, and melt index I₁₀ is measured according to ASTM D1238at 190° C. and 10 kg load. In other embodiments the melt flow ratio(I₁₀/I₂) is from 5 to 50, in others, the melt flow ratio is from 5 to25, in others, the melt flow ratio is from 5 to 9.

In some embodiments, the long-chain branched polymer resulting from thelong-chain polymerization process may have a molecular-weightdistribution (MWD) from 1 to 20, where MWD is defined as M_(w)/M_(n)with M_(w) being a weight average molecular weight and M_(n) being anumber average molecular weight. In other embodiments, the polymersresulting from the catalyst system have a MWD from 1 to 10. Anotherembodiment includes a MWD from 1 to 3; and other embodiments include MWDfrom 1.5 to 2.5.

Parallel Polymerization Reactor (PPR)

The small scale solution polymerization examples are performed in 15 mLvials using a total liquid volume of 5 mL, a constant ethylene pressureof 150 psig, and a polymerization temperature of 120° C. The 5 mL liquidvolume consists of a 0.84 mL comonomer mixture containing 500 nmolMMAO-3A, catalyst and activator solution in toluene, with sufficientIsopar-E added to achieve a 5 mL liquid volume. Hydrogen (H₂) was addedto the reaction mixture by simultaneously pre-pressurizing the emptyreaction vials with 20±3 psig H₂ at 80° C., such that experiments forany given diene are performed with the same H₂ load. All liquid volumeswere dispensed at room temperature and added volumetrically in relationto the 5 mL total volume. The catalyst was added last to the reactionmixture as a 5 mM solution in toluene which was separately activated by1.5 equivalents of Co-Catalyst A (methyldi(tetradecyl)ammoniumtetrakis(pentafluorophenyl)borate). The comonomer solution was composedpredominantly of 1-octene with a minor (0-6%) volume fraction of a dienespecies. The polymerizations were run for times not exceeding about 30minutes and were quenched by CO addition followed by vialde-pressurization.

Gel Permeation Chromatography (GPC) (Conventional GPC)

The chromatographic system consisted of a PolymerChar GPC-IR (Valencia,Spain) high temperature GPC chromatograph equipped with an internal IR5infra-red detector (IR5) and 4-capillary viscometer (DV) coupled to aPrecision Detectors (Now Agilent Technologies) 2-angle laser lightscattering (LS) detector Model 2040. For all absolute Light scatteringmeasurements, the 15 degree angle is used for measurement. Theautosampler oven compartment was set at 1600 Celsius and the columncompartment was set at 1500 Celsius. The columns used were 4 Agilent“Mixed A” 30 cm 20-micron linear mixed-bed columns. The chromatographicsolvent used was 1,2,4 trichlorobenzene and contained 200 ppm ofbutylated hydroxytoluene (BHT). The solvent source was nitrogen sparged.The injection volume used was 200 microliters and the flow rate was 1.0milliliters/minute.

Calibration of the GPC column set was performed with at least 20 narrowmolecular weight distribution polystyrene standards with molecularweights ranging from 580 to 8,400,000 and were arranged in 6 “cocktail”mixtures with at least a decade of separation between individualmolecular weights. The standards were purchased from AgilentTechnologies. The polystyrene standards were prepared at 0.025 grams in50 milliliters of solvent for molecular weights equal to or greater than1,000,000, and 0.05 grams in 50 milliliters of solvent for molecularweights less than 1,000,000. The polystyrene standards were dissolved at80 degrees Celsius with gentle agitation for 30 minutes. The polystyrenestandard peak molecular weights were converted to polyethylene molecularweights using Equation 52 (as described in Williams and Ward, J. Polym.Sci., Polym. Let., 6, 621 (1968)):

M _(polyethylene) =A×(M _(polystyrene))^(B)  (52)

where M is the molecular weight, A has a value of 0.4315 and B is equalto 1.0.

A polynomial between 3^(rd) and 5^(th) order was used to fit therespective polyethylene-equivalent calibration points. A smalladjustment to A (from approximately 0.415 to 0.44) was made to correctfor column resolution and band-broadening effects such that NISTstandard NBS 1475 is obtained at 52,000 Mw.

The total plate count of the GPC column set was performed with Eicosane(prepared at 0.04 g in 50 milliliters of TCB and dissolved for 20minutes with gentle agitation). The plate count (Equation 53) andsymmetry (Equation 54) were measured on a 200 microliter injectionaccording to the following equations:

$\begin{matrix}{{{Plate}\mspace{14mu}{Count}} = {5.54*\left( \frac{\left( {RV_{PeakMax}} \right.}{{Peak}\mspace{14mu}{Width}\mspace{14mu}{at}\mspace{14mu}\frac{1}{2}{height}} \right)^{2}}} & (53)\end{matrix}$

where RV is the retention volume in milliliters, the peak width is inmilliliters, the peak max is the maximum height of the peak, and ½height is ½ height of the peak maximum.

$\begin{matrix}{{Symmetry} = \frac{\left( {{{Rear}\mspace{14mu}{Peak}\mspace{14mu}{RV}_{onete{nthheight}}} - {RV_{Peakmax}}} \right)}{\left( {{RV_{Peakmax}} - {{Front}\mspace{14mu}{Peak}\mspace{14mu}{RV}_{onete{nthheight}}}} \right)}} & (54)\end{matrix}$

where RV is the retention volume in milliliters and the peak width is inmilliliters, Peak max is the maximum position of the peak, one tenthheight is 1/10 height of the peak maximum, and where rear peak refers tothe peak tail at later retention volumes than the peak max and wherefront peak refers to the peak front at earlier retention volumes thanthe peak max. The plate count for the chromatographic system should begreater than 24,000 and symmetry should be between 0.98 and 1.22.

Samples were prepared in a semi-automatic manner with the PolymerChar“Instrument Control” Software, wherein the samples were weight-targetedat 2 mg/ml, and the solvent (contained 200 ppm BHT) was added to a prenitrogen-sparged septa-capped vial, via the PolymerChar high temperatureautosampler. The samples were dissolved for 2 hours at 1600 Celsiusunder “low speed” shaking.

The calculations of M_(n(GPC)), M_(w(GPC)), and M_(z(GPC)) were based onGPC results using the internal IR5 detector (measurement channel) of thePolymerChar GPC-IR chromatograph according to Equations 55-57, usingPolymerChar GPCOne™ software, the baseline-subtracted IR chromatogram ateach equally-spaced data collection point (i), and the polyethyleneequivalent molecular weight obtained from the narrow standardcalibration curve for the point (i).

$\begin{matrix}{{Mn}_{({GPC})} = \frac{\sum\limits^{i}{IR}_{i}}{\sum\limits^{i}\left( {{IR}_{i}\text{/}M_{{polyethylene}_{i}}} \right)}} & (55) \\{{Mw}_{({GPC})} = \frac{\sum\limits^{i}\left( {{IR}_{i}*M_{{polyethylene}_{i}}} \right)}{\sum\limits^{i}{IR}_{i}}} & (56) \\{{Mz}_{({GPC})} = \frac{\sum\limits^{i}\left( {{IR}_{i}*M_{{polyethylene}_{i}}^{2}} \right)}{\sum\limits^{i}\left( {{IR}_{i}*M_{{polyethylene}_{i}}} \right)}} & (57)\end{matrix}$

In order to monitor the deviations over time, a flowrate marker (decane)was introduced into each sample via a micropump controlled with thePolymerChar GPC-IR system. This flowrate marker (FM) was used tolinearly correct the pump flowrate (Flowrate_((nominal))) for eachsample by RV alignment of the respective decane peak within the sample(RV_((FM Sample))) to that of the decane peak within the narrowstandards calibration (RV_((FM Calibrated))). Any changes in the time ofthe decane marker peak are then assumed to be related to a linear-shiftin flowrate (Flowrate_((effective))) for the entire run. To facilitatethe highest accuracy of a RV measurement of the flow marker peak, aleast-squares fitting routine is used to fit the peak of the flow markerconcentration chromatogram to a quadratic equation. The first derivativeof the quadratic equation is then used to solve for the true peakposition. After calibrating the system based on a flow marker peak, theeffective flowrate (with respect to the narrow standards calibration) iscalculated as Equation 58. Processing of the flow marker peak was donevia the PolymerChar GPCOne™ Software. Acceptable flowrate correction issuch that the effective flowrate should be within +/−2% of the nominalflowrate.

Flowrate_((effective))=Flowrate_((nominal))*(RV _((FM Calibrated)) /RV_((FM Sample)))  (58)

Triple Detector GPC (TDGPC) (Absolute GPC)

The chromatographic system, run conditions, column set, columncalibration and calculation conventional molecular weight moments andthe distribution were performed according to the method described in GelPermeation Chromatography (GPC).

For the determination of the viscometer and light scattering detectoroffsets from the IR5 detector, the Systematic Approach for thedetermination of multi-detector offsets is done in a manner consistentwith that published by Balke, Mourey, et. al. (Mourey and Balke,Chromatography Polym. Chpt 12, (1992)) (Balke, Thitiratsakul, Lew,Cheung, Mourey, Chromatography Polym. Chpt 13, (1992)), optimizingtriple detector log (MW and IV) results from a broad homopolymerpolyethylene standard (M_(w)/M_(n)>3) to the narrow standard columncalibration results from the narrow standards calibration curve usingPolymerChar GPCOne™ Software.

The absolute molecular weight data is obtained in a manner consistentwith that published by Zimm (Zimm, B. H., J. Chem. Phys., 16, 1099(1948)) and Kratochvil (Kratochvil, P., Classical Light Scattering fromPolymer Solutions, Elsevier, Oxford, N.Y. (1987)) using PolymerCharGPCOne™ software. The overall injected concentration, used in thedetermination of the molecular weight, is obtained from the massdetector area and the mass detector constant, derived from a suitablelinear polyethylene homopolymer, or one of the polyethylene standards ofknown weight average molecular weight. The calculated molecular weights(using GPCOne™) are obtained using a light scattering constant, derivedfrom one or more of the polyethylene standards mentioned below, and arefractive index concentration coefficient, dn/dc, of 0.104. Generally,the mass detector response (IR5) and the light scattering constant(determined using GPCOne™) may be determined from a linear standard witha molecular weight in excess of about 50,000 g/mole. The viscometercalibration (determined using GPCOne™) may be accomplished using themethods described by the manufacturer, or, alternatively, by using thepublished values of suitable linear standards, such as StandardReference Materials (SRM) 1475a (available from National Institute ofStandards and Technology (NIST)). A viscometer constant (obtained usingGPCOne™) is calculated which relates specific viscosity area (DV) andinjected mass for the calibration standard to its intrinsic viscosity.The chromatographic concentrations are assumed low enough to eliminateaddressing 2nd viral coefficient effects (concentration effects onmolecular weight).

The absolute weight average molecular weight (M_(w(Abs))) is obtained(using GPCOne™) from the Area of the Light Scattering (LS) integratedchromatogram (factored by the light scattering constant) divided by themass recovered from the mass constant and the mass detector (IR5) area.The molecular weight and intrinsic viscosity responses are linearlyextrapolated at chromatographic ends where signal to noise becomes low(using GPCOne™). Other respective moments, M_(n(Abs)) and M_(z(Abs)) arebe calculated according to equations 59-60 as follows:

$\begin{matrix}{{Mn}_{({Abs})} = \frac{\sum\limits^{i}{IR}_{i}}{\sum\limits^{i}\left( {{IR}_{i}\text{/}M_{{Absolute}_{i}}} \right)}} & (59) \\{{Mz}_{({Abs})} = \frac{\sum\limits^{i}\left( {{IR}_{i}*M_{{Absolute}_{i}}^{2}} \right)}{\sum\limits^{i}\left( {{IR}_{i}*M_{{Absolute}_{i}}} \right)}} & (60)\end{matrix}$

g′_(ave) Values

g′ is defined as the viscosity of a branched polymer divided by theviscosity of a linear polymer at the same MW:

$\begin{matrix}{g^{\prime} = {\frac{\lbrack\eta\rbrack_{branched}}{\lbrack\eta\rbrack_{linear}}❘_{{same}\mspace{11mu} M}}} & (61)\end{matrix}$

g′_(ave) or average g′ is the weight-averaged value of g′ (B. H. Zimm,W. H. Stockmayer, J. Chem. Phys. 1949, 17, 1301).

Dynamic Mechanical Spectrum (or Small Angle Oscillatory Shear)

The complex viscosity (η*), moduli (G′, G″), tan delta, and phase angle(δ) are obtained by dynamic oscillatory frequency sweep test in afrequency range from 0.1 to 100 rad/s, at 190° C. The level of strain isset within the linear viscoelastic regime as identify by a strain sweeptest at 100 rad/s at 190° C. Tests are performed with stainless steelparallel plates of 25 mm diameter on a strain controlled rheometerARES-G2 by TA Instruments. Samples of 3.3 mm thickness are squeezed andthen trimmed in two steps prior to the actual test. In the first step,the sample are allowed to melt for 2.5 min, squeezed to 3 mm gap andtrimmed. After an additional 2.5 min of soak time at 190° C., the sampleare squeezed to 2 mm gap, and the excess of material trimmed. The methodhas an additional five minute delay built in to allow the system toreach thermal equilibrium. Tests are performed under nitrogenatmosphere.

gpcBR Branching Index by Triple Detector GPC (TDGPC)

The gpcBR branching index was determined by first calibrating the lightscattering, viscosity, and concentration detectors as describedpreviously. Baselines were then subtracted from the light scattering,viscometer, and concentration chromatograms. Integration windows werethen set, to ensure integration of all of the low molecular weightretention volume range in the light scattering and viscometerchromatograms that indicate the presence of detectable polymer from therefractive index chromatogram. Linear polyethylene standards were thenused to establish polyethylene and polystyrene Mark-Houwink constants.Upon obtaining the constants, the two values were used to construct twolinear reference conventional calibrations for polyethylene molecularweight and polyethylene intrinsic viscosity as a function of elutionvolume, as shown in Equations (62) and (63):

$\begin{matrix}{M_{PE} = {\left( \frac{K_{PS}}{K_{PE}} \right)^{{1/\alpha_{PE}} + 1} \cdot M_{PS}^{\alpha_{PS} + {1/\alpha_{PE}} +}}} & (62) \\{\lbrack\eta\rbrack_{PE} = {{K_{PS} \cdot M_{PS}^{a + 1}}\text{/}M_{PE}}} & (63)\end{matrix}$

The gpcBR branching index is a robust method for the characterization oflong chain branching as described in Yau, Wallace W., “Examples of Using3D-GPC-TREF for Poly-olefin Characterization,” Macromol. Symp., 2007,257, 29-45. The index avoids the “slice-by-slice” TDGPC calculationstraditionally used in the determination of g′ values and branchingfrequency calculations, in favor of whole polymer detector areas. FromTDGPC data, one can obtain the sample bulk absolute weight averagemolecular weight (M_(w), abs) by the light scattering (LS) detector,using the peak area method. The method avoids the “slice-by-slice” ratioof light scattering detector signal over the concentration detectorsignal, as required in a traditional g′ determination. With TDGPC,sample intrinsic viscosities were also obtained independently usingEquation (64). The area calculation in this case offers more precision,because, as an overall sample area, it is much less sensitive tovariation caused by detector noise and TDGPC settings on baseline andintegration limits. More importantly, the peak area calculation was notaffected by the detector volume offsets. Similarly, the high-precision,sample intrinsic viscosity (IV) was obtained by the area method inEquation (64):

$\begin{matrix}{{{IV} = {\lbrack\eta\rbrack = {{\sum\limits_{i}{w_{i}{IV}_{i}}} = {{\sum\limits_{i}{\left( \frac{C_{i}}{\sum\limits_{i}C_{i}} \right){IV}_{i}}} = {\frac{\sum\limits_{i}{C_{i}IV_{i}}}{\sum\limits_{i}C_{i}} = {\frac{\sum\limits_{i}{DP_{i}}}{\sum\limits_{i}C_{i}} = \frac{{DP}\mspace{14mu}{Area}}{{Conc}.\mspace{14mu}{Area}}}}}}}},} & (64)\end{matrix}$

In Equation (64), DPi stands for the differential pressure signalmonitored directly from the online viscometer. To determine the gpcBRbranching index, the light scattering elution area for the samplepolymer was used to determine the molecular weight of the sample. Theviscosity detector elution area for the sample polymer was used todetermine the intrinsic viscosity (IV or [η]) of the sample. Initially,the molecular weight and intrinsic viscosity for a linear polyethylenestandard sample, such as SRM1475a or an equivalent, were determinedusing the conventional calibrations (“cc”) for both molecular weight andintrinsic viscosity as a function of elution volume:

$\begin{matrix}{\lbrack\eta\rbrack_{CC} = {{\sum\limits_{i}{\left( \frac{C_{i}}{\sum\limits_{i}C_{i}} \right){IV}_{i}}} = {\sum\limits_{i}{w_{i}I{V_{{cc},i}.}}}}} & (65)\end{matrix}$

Equation (66) was used to determine the gpcBR branching index:

$\begin{matrix}{{{gpcBR} = \left\lbrack {{\left( \frac{\lbrack\eta\rbrack_{CC}}{\lbrack\eta\rbrack} \right) \cdot \left( \frac{M_{W}}{M_{W,{CC}}} \right)^{\alpha_{PE}}} - 1} \right\rbrack},} & (66)\end{matrix}$

wherein [η] is the measured intrinsic viscosity, [η]_(cc) is theintrinsic viscosity from the conventional calibration (or conv GPC), Mwis the measured weight average molecular weight, and M_(w,cc) is theweight average molecular weight of the conventional calibration. Theweight average molecular weight by light scattering (LS) is commonlyreferred to as “absolute weight average molecular weight” or“M_(w)(abs).” The M_(w,cc) from using conventional GPC molecular weightcalibration curve (“conventional calibration”) is often referred to as“polymer chain backbone molecular weight,” “conventional weight averagemolecular weight” and “M_(w)(conv).”

All statistical values with the “cc or conv” subscript are determinedusing their respective elution volumes, the corresponding conventionalcalibration as previously described, and the concentration (Ci). Thenon-subscripted values are measured values based on the mass detector,LALLS, and viscometer areas. The value of K_(PE) is adjustediteratively, until the linear reference sample has a gpcBR measuredvalue of zero. For example, the final values for a and Log K for thedetermination of gpcBR in this particular case are 0.725 and −3.355,respectively, for polyethylene, and 0.722 and −3.993, respectively, forpolystyrene. Once the K and α values have been determined using theprocedure discussed.

Previously, the procedure was repeated using the branched samples. Thebranched samples were analyzed using the final Mark-Houwink constants asthe best “cc” calibration values.

The interpretation of gpcBR is straight forward. For linear polymers,gpcBR will be close to zero, since the values measured by LS andviscometry will be close to the conventional calibration standard. Forbranched polymers, gpcBR will be higher than zero, especially with highlevels of long chain branching, because the measured polymer molecularweight will be higher than the calculated M_(w,cc), and the calculatedIV_(cc) will be higher than the measured polymer IV. In fact, the gpcBRvalue represents the fractional IV change due to the molecular sizecontraction effect as a result of polymer branching. A gpcBR value of0.5 or 2.0 would mean a molecular size contraction effect of IV at thelevel of 50% and 200%, respectively, versus a linear polymer molecule ofequivalent weight. For these particular examples, the advantage of usinggpcBR, in comparison to a traditional “g′ index” and branching frequencycalculations, is due to the higher precision of gpcBR. All of theparameters used in the gpcBR index determination are obtained with goodprecision, and are not detrimentally affected by the low TDGPC detectorresponse at high molecular weight from the concentration detector.Errors in detector volume alignment also do not affect the precision ofthe gpcBR index determination.

Batch Reactor Polymerization Procedure

The batch reactor polymerization reactions are conducted in a 2 L Parr™batch reactor. The reactor is heated by an electrical heating mantle,and is cooled by an internal serpentine cooling coil containing coolingwater. Both the reactor and the heating/cooling system are controlledand monitored by a Camile™ TG process computer. The bottom of thereactor is fitted with a dump valve that empties the reactor contentsinto a stainless steel dump pot. The dump pot is prefilled with acatalyst kill solution (typically 5 mL of an Irgafos/Irganox/toluenemixture). The dump pot is vented to a 30 gallon blow-down tank, withboth the pot and the tank purged with nitrogen. All solvents used forpolymerization or catalyst makeup are run through solvent purificationcolumns to remove any impurities that may affect polymerization. The1-octene and IsoparE are passed through two columns, the firstcontaining A2 alumina, the second containing Q5. The ethylene is passedthrough two columns, the first containing A204 alumina and 4{acute over(Å)} molecular sieves, the second containing Q5 reactant. The N₂, usedfor transfers, is passed through a single column containing A204alumina, 4{acute over (Å)} molecular sieves and Q5.

The reactor is loaded first from the shot tank that may contain IsoparEsolvent and/or 1-octene, depending on reactor load. The shot tank isfilled to the load set points by use of a lab scale to which the shottank is mounted. After liquid feed addition, the reactor is heated up tothe polymerization temperature set point. If ethylene is used, it isadded to the reactor when the ethylene is at the reaction temperature tomaintain reaction pressure set point. The amount of ethylene added ismonitored by a micro-motion flow meter (Micro Motion). For someexperiments, the standard conditions at 150° C. are 13 g ethylene, 15 g1-octene, 240 psi hydrogen in 585 g of IsoparE, and the standardconditions at 150° C. are 15 g ethylene, 45 g 1-octene, 200 psi hydrogenin 555 g of IsoparE.

The procatalyst and activators are mixed with the appropriate amount ofpurified toluene to achieve a desired molarity solution. The procatalystand activators are handled in an inert glove box, drawn into a syringeand pressure transferred into the catalyst shot tank. The syringe isrinsed three times with 5 mL of toluene. Immediately after the catalystis added, the run timer begins. If ethylene is used, it is added by theCamile to maintain reaction pressure set point in the reactor. Thepolymerization reactions are run for 10 minutes, then the agitator isstopped, and the bottom dump valve is opened to empty reactor contentsto the dump pot. The contents of the dump pot are poured into trays andplaced in a lab hood where the solvent was evaporated off overnight. Thetrays containing the remaining polymer are transferred to a vacuum oven,where they are heated up to 140° C. under vacuum to remove any remainingsolvent. After the trays cool to ambient temperature, the polymers wereweighed for yield to measure efficiencies, and submitted for polymertesting.

EXAMPLES Tetra-Functional Branching in the Presence of VariousMulti-Chain Catalysts and Various Dienes

The results of the small scale polymerizations are summarized in Table 3through Table 7 (experimental is in the parallel polymerizationreactors, PPR). The polymer results recorded in Table 3 through Table 7were produced by polymerizing ethylene, octene, and a diene species inthe presence of multi-chain catalysts and single chain catalystcontrols. The polymer results in each table of the Table 3 through Table7 were the product of various catalysts and diene species. The resultsin Table 3 are based on the polymer products of 3-methyl-1,4-pentadiene,ethylene, and octene in the presence of Comparative Catalyst C1 (“Comp.Cat. C1”), Catalyst 1 (“Cat. 1”), and Catalyst 2 (“Cat. 2”). The resultsin Table 4 are based on the polymer products of 1,4-pentadiene,ethylene, and octene in the presence of Cat. 2 and Catalyst 4 (“Cat.4”). The results in Table 5 are based on the polymer products of1,5-hexadiene, ethylene, and octene in the presence of Comp. Cat. C1,Catalyst 3 (“Cat. 3”), Catalyst 5 (“Cat. 5”), and Catalyst 6 (“Cat. 6”).The results in Table 6 are based on the polymer products of1,7-octadiene, ethylene, and octene in the presence of Comp. Cat. C1,Cat. 6, Cat. 2, and Cat. 4. The results in Table 7 are based on thepolymer products of 1,9-decadiene, ethylene, and octene in the presenceof Cat. 3, Cat. 5, Cat. 6, and Cat. 2 (Figueroa, R.; Froese, R. D.; He,Y.; Klosin, J.; Theriault, C. N.; Abboud, K. A. Organometallics 2011,30, 1695-1709, Froese, R. D.; Jazdzewski, B. A.; Klosin, J.; Kuhlman, R.L.; Theriault, C. N.; Welsh, D. M; Abboud, K. A. Organometallics 2011,30, 251-262)

The single chain catalyst in Series 3.C (Comp. Cat. C1) incorporatedincreased amount of α-olefin as indicated by the two-fold and higheroctene level in the polymer when compared to the other catalysts. Whenusing the single chain catalyst in Series 3.C ((Comp. Cat. C1) thevarious levels of added diene had no significant effect on the polymerMWD. However, adding the diene to the dual chain catalysts in Table 3through Table 7 resulted in higher values of M_(w) and M_(p) as dienelevel was increased, and often there was no evidence of a high molecularweight tail forming.

In each example contain diene, the amount of diene incorporated into thereactor was small in comparison to the other reactants in the reactor.Therefore, the addition of diene did not affect the amount of comonomer,ethylene, and solvent added into the reactor.

Example 1—Tetra-Functional Branching with 3-methyl-1,4-pentadiene

TABLE 3 Small scale polymerizations (PPR) with 3-methyl-1,4-pentadieneas the diene species. Diene Cat in Octene Poly Conventional GPC Data andMetrics Added Octene in Poly Yield Time Mn Mw Mp Mw / Mp / Ex. Cat(μmol) (vol %) (mol %) (g) (s) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL)3.C.1 Comp. 0.020 0 37.4 0.13 1801 3,049 7,138 6,607 1.00 1.00 0.0720.021 3.C.2 Cat C1 4 35.7 0.14 1801 3,282 7,515 7,079 1.05 1.07 0.0570.021 3.C.3 6 31.5 0.18 116 3,030 6,883 6,457 0.96 0.98 0.063 0.0203.1.1 Cat. 1 0.025 0 16.6 0.28 37 6,471 35,214 14,454 1.00 1.00 0.410.12 3.1.2 2 15.3 0.24 37 6,608 41,256 15,849 1.17 1.10 0.055 0.0693.1.3 4 13.4 0.18 43 7,797 53,276 21,878 1.51 1.51 0.12 0.034 3.1.4 613.1 0.18 42 9,310 55,406 25,119 1.57 1.74 0.26 0.059 3.4.1 Cat. 2 0.0250 15.2 0.20 28 5,293 27,847 11,749 1.00 1.00 0.203 0.083 3.4.2 2 12.50.18 32 6,921 41,491 17,783 1.49 1.51 −0.29 0.016 3.4.3 4 11.6 0.16 458,142 46,098 22,388 1.66 1.91 0.083 0.028 3.4.4 6 12.0 0.17 38 8,33344,892 21,878 1.61 1.86 0.15 0.037

Example 2—Tetra-Functional Branching with 1,4-pentadiene

TABLE 4 Small scale polymerizations (PPR) with 1,4-pentadiene as thediene species. Diene Cat in Octene Poly Conventional GPC Data andMetrics Added Octene in Poly Yield Time Mn Mw Mp Mw / Mp / Ex. Cat μmolvol % mol % g (s) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 4.3.1 Cat. 20.015 0 12.9 0.17 30 7,330 29,587 16,596 1.00 1.00 0.13 0.033 4.3.2 112.6 0.16 40 9,251 46,581 28,184 1.57 1.70 0.39 0.063 4.3.3 2 11.3 0.1547 12,460 68,277 45,710 2.31 2.75 0.31 0.06 4.3.4 3 9.8 0.15 39 15,99982,060 53,705 2.77 3.24 0.13 0.057 P2.4.1 Cat. 4 0.020 0 6.3 0.08 1125,613 18,813 13,804 1.00 1.00 −0.10 0.02 P2.4.2 1 5.9 0.08 202 6,72622,032 16,218 1.17 1.17 −0.089 0.024 P2.4.3 2 5.6 0.07 254 6,983 24,00317,783 1.28 1.29 −0.063 0.023 4.4.4 3 5.5 0.07 263 7,815 26,562 20,8931.41 1.51 0.055 0.031 Diene Cat in Octene Poly Absolute GPC Data andMetrics Added Octene in Poly Yield Time Mn Mw Mp Mw / Mp / Ex. Cat(μmol) (vol %) (mol %) (g) (s) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL)4.4.1 Cat. 4 0.020 0 6.3 0.08 112 5,383 18,505 11,220 1.00 1.00 −0.310.026 4.4.2 1 5.9 0.08 202 5,752 22,399 15,488 1.21 1.38 0.14 0.0354.4.3 2 5.6 0.07 254 6,785 26,508 20,418 1.43 1.82 0.23 0.043 4.4.4 35.5 0.07 263 6,444 27,369 26,916 1.48 2.40 0.22 0.036

FIG. 20 illustrates the shift in peak weight average molecular weight asthe amount of diene is increased. In FIG. 20, the P2.4.1-P2.4.4 series,as recorded in Table 4, are plotted as dWd Log M as a function as Log M,which is a GPC plot. As the volume percent of diene increased, the peakin the GPC plot shifted right.

Example 3—Tetra-Functional Branching with 1,5-hexadiene

TABLE 5 Small scale polymerizations (PPR) with 1,5-hexadiene as thediene species. . Diene Cat in Octene Poly Conventional GPC Data andMetrics Added Octene in Poly Yield Time Mn Mw Mp Mw/ Mp/ Ex Cat (μmole)(vol %) (mol %) (g) (s) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 5.C.1 Comp0.020 0 34.3 0.09 1801  3,394  7,576  6,761 1.00 1.00 0.01 0.018 5.C.2Cat. 1 34.4 0.11 1802  3,550  7,596  6,457 1.00 0.95 −0.006 0.018 5.C.3C1 2 35.5 0.10 1801  3,742  8,000  7,244 1.06 1.07 0.011 0.020 5.C.4 330.3 0.11 1800  3,509  8,059  7,244 1.06 1.07 0.001 0.021 5.1.1 Cat. 30.012 0 7.5 0.09 104 19,503 42,284 38,020 1.00 1.00 0.095 0.025 5.1.2 16.3 0.05 1800 22,832 48,933 42,659 1.16 1.12 0.12 0.022 5.1.3 2 8.6 0.061801 23,497 51,097 45,710 1.21 1.20 0.11 0.024 5.2.4 3 6.4 0.03 180024,880 54,297 48,979 1.28 1.29 0.10 0.024 6.2.1 Cat. 5 0.012 0 6.6 0.0862  4,345 11,676 10,000 1.00 1.00 0.078 0.021 6.2.2 1 6.3 0.08 81  6,21215,666 13,183 1.34 1.32 0.065 0.022 6.2.3 2 5.4 0.07 159  9,304 22,29019,055 1.91 1.91 0.036 0.022 6.2.4 3 8.9 0.07 118  7,337 17,426 15,8491.49 1.58 −0.007 0.019 6.3.1 Cat. 6 0.015 0 11.6 0.09 120  6,030 18,08515,488 1.00 1.00 0.012 0.020 6.3.2 1 11.5 0.09 117  6,801 18,582 16,5961.03 1.07 0.043 0.022 6.3.3 2 9.3 0.07 1103  8,275 24,908 21,380 1.381.38 −0.002 0.025 6.3.4 3 10.4 0.09 135  9,283 33,604 28,841 1.86 1.860.092 0.028

Example 4—Tetra-Functional Branching with 1,7-octadiene

TABLE 6 Small scale polymerizations (PPR) with 1,7-octadiene as thediene species. Diene Cat in Octene Poly Conventional GPC Data andMetrics Added Octene in Poly Yield Time Mn Mw Mp Mw/ Mp/ Ex. Cat (μmole)(vol %) (mol %) (g) (s) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 6.C.1Comp. 0.020 0 37 0.24 48  1,127   2,230  1,862 1.00 1.00 0.039 0.0196.C.2 Cat. C1 2 38 0.24 51  1,125   2,248  1,905 1.01 1.02 0.006 0.0226.C.3 4 37 0.23 56  1,124   2,201  1,862 0.99 1.00 0.019 0.020 6.C.4 631 0.24 57  1,135   2,352  1,995 1.05 1.07 0.028 0.021 6.3.1 Cat. 60.015 0 13 0.10 62  4,825  13,681 12,023 1.00 1.00 0.10 0.024 6.3.2 2 120.10 92  5,755  21,647 16,218 1.58 1.35 −0.17 0.023 6.3.3 4 11 0.10 111 6,913  31,963 30,200 2.34 2.51 0.15 0.040 6.3.4 6 10 0.10 125  6,870 54,771 39,812 4.00 3.31 0.19 0.074 6.4.1 Cat. 2 0.015 0 11 0.14 3812,882  41,563 28,184 1.00 1.00 0.080 0.029 6.4.2 2 11 0.15 34 12,623 61,652 33,885 1.48 1.20 0.36 0.061 6.4.3 4 10 0.14 43 17,699 101,92147,865 2.45 1.70 0.19 0.094 6.5.1 Cat. 4 0.020 0 6 0.07 131  7,927 20,744 15,849 1.00 1.00 −0.089 0.019 6.5.2 2 6 0.08 234  6,838  22,36815,488 1.08 0.98 −0.073 0.025 6.5.3 4 6 0.07 267  7,554  24,192 15,4881.17 0.98 −0.032 0.028 6.5.4 6 5 0.07 353  7,459  25,772 16,218 1.241.02 −0.058 0.025

Example 5—Tetra-Functional Branching with 1,9-decadiene

TABLE 7 Small scale polymerizations (PPR) with 1,9-decadiene as thediene species. Diene Cat in Octene Poly Conventional GPC Data andMetrics Added Octene in Poly Yield Time Mn Mw Mp Mw/ Mp/ Ex. Cat (μmol)(vol %) (mol %) (g) (s) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 7.1.1 Cat.3 0.012 0 11 0.13 50 10,690  25,915 20,418 1.00 1.00 0.14 0.027 7.1.2 29 0.14 45 10,420  43,497 26,916 1.68 1.32 0.23 0.044 7.1.3 4 7 0.11 5222,410  93,485 54,956 3.61 2.69 0.29 0.079 7.2.1 Cat. 5 0.012 0 — 0.1132  2,961  10,454  7,413 1.00 1.00 −0.012 0.024 7.2.2 2 8 0.11 39  4,151 14,343  8,912 1.37 1.20 0.001 0.027 7.2.3 4 8 0.11 32  3,837  16,98011,220 1.62 1.51 −0.14 0.021 7.2.4 6 6 0.11 32  5,604  22,682 12,8832.17 1.74 −0.19 0.022 7.3.1 Cat. 6 0.015 0 14 0.11 49  5,141  16,27214,125 1.00 1.00 0.092 0.026 7.3.2 2 13 0.11 55  6,537  26,292 18,1971.62 1.29 −0.046 0.033 7.3.3 4 11 0.11 48  8,273  44,729 33,114 2.752.34 0.22 0.062 7.3.4 6 11 0.11 68  9,422  93,485 41,688 5.75 2.95 0.310.15 7.4.1 Cat. 2 0.015 0 10 0.14 36 14,989  47,160 35,482 1.00 1.000.15 0.035 7.4.2 2 10 0.16 33 15,195  87,298 39,812 1.85 1.12 0.0270.073 7.4.3 4 9 0.14 45 21,234 189,596 56,236 4.02 1.58 −0.44 0.0087.4.4 6 9 0.16 39 19,657 301,779 48,979 6.40 1.38 −1.65 0.019 7.5.1 Cat.4 0.020 0 5 0.07 122  9,775  25,249 19,055 1.00 1.00 −0.072 0.020 7.5.22 5 0.08 146  8,156  27,499 18,197 1.09 0.95 −0.006 0.030 7.5.3 4 5 0.07140  9,137  31,407 18,197 1.24 0.95 −0.039 0.026 7.5.4 6 5 0.07 140 7,832  34,706 17,783 1.37 0.93 0.001 0.037

BRANCHED EXAMPLES FROM BATCH REACTOR

The molecular weight distribution (MWD) curve and DSC of two branchedexamples were studied and compared to linear samples.

Batch Reactor Example 1

In Table 8 to Table 12, the polymer characteristic of a comparativelinear polymer sample (1C) was compared to branched polymers from abatch reactor. Polymerizations reactions occurred at a temperature of150° C., in 555 g of ISOPAR-E™ and a hydrogen pressure (ΔH₂) of 200 psi.The ethylene pressure was held constant at 150 psi in the presences of0.3 μmole of Catalyst 8, 0.36 μmole of Co-Catalyst A(methyldi(tetradecyl)ammonium tetrakis(pentafluorophenyl)borate), and 10μmole MMAO-3A.

TABLE 8 Polymer Characteristics of the batch reactor polymer of Example1 and the comparative. Diene Conventional GPC Data and Metrics AddedYield Octene T_(m) Mn Mw Mp Ex. Diene (g) (g) (mol %) (° C.) (g/mole)G_((79/29)) A_(TAIL) 8.C none 0.00 8.7 7.4 78.1 18,160 62,030 54,8200.10 0.026 8.1 1,9-decadiene 0.30 37.1 6.5 80.1 19,740 91,498 66,069−0.06 0.019 Absolute GPC Data and Metrics Mn Mw Mp Ex. Diene (g/mole)G_((79/29)) A_(TAIL) gpcBR 8.C none 21,385  69,811 61,662  0.04 0.0220.34 8.1 1,9-decadiene 23,947 116,322 79,436 −0.05 0.021 0.56

FIG. 21 is a conventional molecular weight distribution curve of thepolymer in series 8.C (linear) and 8.1 (branched), as determined by GPC.The shape of curve of the branched polymer, series 8.1, is altered incomparison to the linear polymer. Additionally, the peak of themolecular weight curve is shifted to the right.

FIG. 22 is an absolute molecular weight distribution curve of thepolymers in series 8.C (linear) and 8.1 (branched), as determined byGPC.

FIG. 23 is the extensional viscosity fixture of branched sample isseries 8.1.

TABLE 9 Dynamic Mechanical Spectrum of branched sample 8.1 at 190° C.Storage Loss Complex Complex Ang freq modulus modulus viscosity Tanmodulus Phase rad/s Pa Pa Pa · s (delta) Pa angle °   0.10 1436 234027457 1.63 2746 58.5   0.16 2067 3057 23285 1.48 3690 55.9   0.25 28843932 19412 1.36 4876 53.7   0.40 3943 5000 15995 1.27 6368 51.7   0.635296 6296 13039 1.19 8227 49.9   1.00 6994 7865 10525 1.12 10525 48.4  1.58 9141 9784 8448 1.07 13390 46.9   2.51 11801 12140 6740 1.03 1693145.8   3.98 15091 15014 5347 0.99 21287 44.9   6.31 19096 18527 42170.97 26606 44.1  10.00 24072 22867 3320 0.95 33202 43.5  15.85 3028428233 2612 0.93 41403 43.0  25.12 38022 34775 2051 0.91 51526 42.4 39.81 47720 42681 1608 0.89 64022 41.8  63.10 59580 51766 1251 0.8778927 41.0 100.00 74626 62604 974 0.84 97408 40.0

The Dynamic Mechanical Spectrum of the branched Example 8.1 was measuredand the results recorded in Table 9. The viscosity at 0.1 radians/secondwas calculated to be 27,457 Pa s and the viscosity at 100 radians/secondwas measured to be 974 Pa s, providing a rheology ratio (V_(0.1)/V₁₀₀)of 28.2.

The elasticity factor m is [((tan (δ_(0.1))−tan(δ₁₀₀))*1000)/(0.1-100))]. The tan (δ_(0.1)) is the tangent of the phaseangle at 0.1 rad/s and the tan (δ₁₀₀) is the tangent of the phase angleat 100 rad/s. The tan (δ_(0.1)) of the branched polymer in Example 8.1was 1.6, and the tan (δ₁₀₀) of Example 1 was 0.8, which yields anelasticity factor of 7.9 at 190° C.

TABLE 10 Dynamic Mechanical Spectrum of the Comparative linear sample8.C at 190° C. Storage Loss Complex Complex Ang freq modulus modulusviscosity Tan modulus Phase rad/s Pa Pa Pa · s (delta) Pa angle°   0.102 89 892 53.30 89 88.9   0.16 3 141 888 48.95 141 88.8   0.25 5 222 88346.32 222 88.8   0.40 9 350 880 40.88 350 88.6   0.63 16 552 875 33.93552 88.3   1.00 33 869 870 26.21 870 87.8   1.58 71 1367 863 19.32 136887.0   2.51 150 2138 853 14.23 2143 86.0   3.98 316 3325 839 10.52 334084.6   6.31 649 5124 819 7.89 5165 82.8  10.00 1294 7798 790 6.02 790480.6  15.85 2502 11700 755 4.68 11964 77.9  25.12 4668 17251 711 3.7017871 74.9  39.81 8351 24781 657 2.97 26151 71.4  63.10 14314 34669 5942.42 37508 67.6 100.00 23447 47067 526 2.01 52584 63.5

The Dynamic Mechanical Spectrum of the comparative 8.C was measured andthe results recorded in Table 10. The shear viscosity at 0.1radians/second was calculated to be 892 Pa s and the shear viscosity at100 radians/second was measured at 526 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 1.7. The tan (δ_(0.1)) of the Comparative linearpolymer, 8.C, was 53.3, and the tan (δ₁₀₀) was 2.0, which yields anelasticity factor of 513.4 at 190° C.

The rheology ratio of the linear comparative polymer resin was very low(1.7) when compared to the rheology ratio of branched Example, series8.1. The increased rheology ratio and the low elasticity factor of thebranched Example 1, series 8.1, are indicative of non-linear polymerbehavior. Strong shear thinning and elastic behavior often exemplifyentangled, long-chain branched polymers.

FIG. 24 is the melt strength obtained by means of a Rheotens device ofthe branched Example 1, series 8.1.

Branched Example 2

In Table 11, branched polyethylene was synthesized where the diene was1,9-decadiene. The branched polymers were polymerized at a temperatureof 150° C., in 555 g of IsoparE and a hydrogen pressure (ΔH₂) of 200psi. The ethylene pressure was held constant at 150 psi in the presencesof 0.3 μmole of Catalyst 7, 0.36 μmole of Co-Catalyst A, and 10 μmoleMMAO-3A.

TABLE 11 Polymer Characteristics of the branched polymer of Example 2and the comparative. Diene Conventional GPC Data and Metrics AbsoluteGPC Data and Metrics Amt Yield Oct. T_(m) Mn Mw Mp Mn Mw Mp Ex. (g) (g)mol % (° C.) (g/mole) G_((79/29)) A_(TAIL) (g/mole) G_((79/29)) A_(TAIL)gpcBR 11.C 0.00 2.7 7.2 81 23,148 61,069 51,286 0.10 0.026 24,297 70,601 58,886 0.02 0.022 0.33 11.1 0.25 52.3 6.8 82 26,464 86,81848,978 −0.04 0.025 32,257 110,354 63,098 −0.08 0.023 0.55

FIG. 25 is a conventional molecular weight distribution curve of thepolymers in Branched Example 2 in series 11.C (linear) and 11.1(branched), as determined by GPC. FIG. 26 is an absolute molecularweight distribution curve of the polymer in series 11.C (linear) and11.1 (branched), as determined by light scattering triple lightdetector. The shape of curve of the branched polymer, series 11.1, isaltered in comparison to the linear polymer.

FIG. 27 is the extensional viscosity obtained by extensional viscosityfixture of the Branched Example 2 in series 11.1.

TABLE 12 Dynamic Mechanical Spectrum of branched Example 2, series 11.1at 190° C. Storage Loss Complex Complex Phase Ang freq modulus modulusviscosity Tan modulus angle rad/s Pa Pa Pa · s (delta) Pa °   0.10 7921577 17643 1.99 1764 63.3   0.16 1185 2125 15351 1.79 2433 60.9   0.251718 2816 13133 1.64 3299 58.6   0.40 2437 3686 11101 1.51 4419 56.5  0.63 3391 4770 9275 1.41 5852 54.6   1.00 4638 6115 7675 1.32 767552.8   1.58 6251 7785 6299 1.25 9984 51.2   2.51 8298 9841 5125 1.1912873 49.9   3.98 10907 12428 4153 1.14 16535 48.7   6.31 14149 156113339 1.10 21069 47.8  10.00 18216 19588 2675 1.08 26749 47.1  15.8523370 24603 2141 1.05 33933 46.5  25.12 29874 30846 1710 1.03 42941 45.9 39.81 38172 38601 1364 1.01 54288 45.3  63.10 48558 47806 1080 0.9868142 44.6 100.00 62045 59123 857 0.95 85704 43.6

The Dynamic Mechanical Spectrum of the comparative was measured and theresults recorded in Table 12. The shear viscosity at 0.1 radians/secondwas calculated to be 17,643 Pa s and the shear viscosity at 100radians/second was measured at 857 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 20.6. The tan (δ_(0.1)) of the branched polymer inExample 2, series 11.1, was 2.0, and the tan (δ₁₀₀) was 1.0, whichyields an elasticity factor of 10.4 at 190° C.

FIG. 28 is the melt strength obtained by a Rheotens device of theBranched Example 2, series 11.1.

Branching Study Under Various Conditions

The tetra-functional “Ladder Branching” was studied under variousconditions, such as increased ethylene pressure, increased octenemonomer, increased starting molecular weight, decreased startingmolecular weight, various dienes, increased or decreased diene amounts,and various multi-chain catalysts.

Example 1: Various Dienes and Amounts of Dienes

The examples in Table 13 to Table 22 were prepared under identicalconditions and polymerized in the presence of Catalyst 7 at atemperature of 150° C. The conditions included: 585 g IsoparE; 15 g1-octene; hydrogen pressure of 240 psi; ethylene pressure of 150 psi;0.3 μmole of Catalyst 7; 0.36 μmole Co-Catalyst A; and 10 μmole MMAO-3A.

TABLE 13 Various dienes tested under identical conditions with Catalyst7 Diene Poly Oct. Conventional GPC Data and Metrics Added Yield PolyT_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) mol % (° C.) (g/mole) Mw_(o)Mp_(o) G_((79/29)) A_(TAIL) 13C none 0.00 8.5 3.4% 109.8 21,059  64,711 51,286 1.00 1.00 0.095 0.027 13.1 1,3- 0.70 5.6 3.0% 110.4 22,523 71,923  58,884 1.11 1.15 0.11 0.027 divinylcyclo pentane 13.2 1,4- 0.357.8 2.9% 112.2 22,400  72,446  52,481 1.12 1.02 0.088 0.027 hexadiene13.3 1,4- 0.75 6.0 2.6% 115.3 22,014  78,338  56,234 1.21 1.10 0.0360.025 hexadiene 13.4 2-me- 0.50 4.1 3.2% 109.6 26,792  83,247  69,1831.29 1.35 0.11 0.026 hexadiene 13.5 2-me- 0.25 8.0 3.0% 110.4 21,430 66,827  50,119 1.03 0.98 0.13 0.030 pentadiene 13.6 2-me- 0.45 10.53.1% 110.8 22,180  65,274  51,286 1.01 1.00 0.14 0.030 pentadiene 13.72-me- 0.50 7.0 2.9% 110.2 21,443  72,911  53,703 1.13 1.05 0.11 0.030pentadiene 13.8 2-me- 1.00 4.7 2.8% 111.6 20,601  75,340  56,234 1.161.10 0.077 0.027 pentadiene 13.9 3-me- 0.25 9.5 3.0% 111.5 22,882 74,859  52,481 1.16 1.02 0.16 0.032 pentadiene 13.10 3-me- 0.50 9.83.2% 111.3 22,261  77,008  50,119 1.19 0.98 0.078 0.027 pentadiene 13.111,9- 0.20 7.4 2.8% 111.2 27,008 100,136  52,481 1.55 1.02 −0.10 0.023decadiene 13.12 1,9- 0.25 7.7 2.6% 115.4 27,458 124,092  63,096 1.921.23 −0.27 0.023 decadiene 13.13 1,9- 0.30 8.0 2.7% 112.9 28,140 125,235 58,884 1.94 1.15 −0.13 0.029 decadiene 13.14 1,9- 0.50 11.3 3.0% 114.222,091 174,694 134,896 2.70 2.63 0.031 0.055 decadiene 13.15 1,5- 0.759.6 3.3% 109.9 22,027  77,375  52,481 1.20 1.02 0.10 0.029 hexadiene13.16 1,8- 0.10 7.9 3.2% 110.2 25,803  91,031  57,544 1.41 1.12 0.0070.024 nonadiene 13.17 1,8- 0.20 6.4 3.4% 108.1 27,604 114,480  81,2831.77 1.58 −0.092 0.019 nonadiene 13.18 1,8- 0.30 3.5 2.9% 104.8 31,057147,084 120,226 2.27 2.34 −0.14 0.031 nonadiene 13.19 1,7- 0.16 9.0 2.9%111.2 26,018  86,041  52,481 1.33 1.02 0.14 0.033 octadiene 13.20 1,7-0.24 9.1 2.7% 112.6 26,711  96,007  53,703 1.48 1.05 −0.002 0.026octadiene 13.21 1,7- 0.30 7.3 2.6% 113.3 28,484 104,803  57,544 1.621.12 −0.14 0.018 octadiene 13.22 1,7- 0.35 8.9 2.8% 112.8 24,548 110,395 58,884 1.71 1.15 −0.15 0.019 octadiene 13.23 1,4- 0.10 7.4 3.7% 111.319,698  81,929  64,565 1.27 1.26 0.16 0.033 pentadiene 13.24 1,4- 0.201.9 3.0% 105.1 31,887 101,861  67,608 1.57 1.32 −0.11 0.013 pentadiene13.25 1,4- 0.30 1.1 2.3% 108.7 29,809 107,172  69,183 1.66 1.35 −0.210.013 pentadiene Absolute GPC Data and Metrics Mn Mw Mp Ex. (g/mole)Mw/Mw_(o) Mp/Mp_(o) G_((79/29)) A_(TAIL) V_(0,1)/V₁₀₀ m gpcBR 13C 25,552 67,491  60,258 1.00 1.00 0.021 0.020 1.5 769 0.18 13.12 36,181 159,527 67,611 2.36 1.12 0.018 0.041 121.4 0.412 0.56 13.16 31,935 105,263 67,611 1.56 1.12 0.082 0.032 28.2 6.338 0.35 13.17 38,628 143,672 97,728 2.13 1.62 0.066 0.027 110.4 1.025 0.51 13.18 42,764 198,061128,832 2.93 2.14 0.043 0.041 — 0.70 13.22 33,516 138,854  70,797 2.061.17 0.048 0.037 101.4 1.018 0.52

The results in Table 13 indicated that when diene was present in thepolymerization reaction, molecular weight increased without a highmolecular weight tail.

Example 2: Conditions that Yield High Molecular Weights

TABLE 14 Nonadiene tested under conditions giving a higher linear MWDiene Poly Octene Conventional GPC Data and Metrics Amt Yield in PolyT_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.) (g/mole) Mw_(o)Mp_(o) G_((79/29)) A_(TAIL) 14.C none 0 12.4 3.2 109.5 31,796 93,45570,795 1.00 1.00 0.13 0.027 14.1 1,8-nonadiene 0.2 6.9 2.9 106.4 46,835211,514 141,254 2.26 2.00 0.03 0.031 14.2 1,8-nonadiene 0.5 3.5 2.8107.1 52,396 387,644 288,403 4.15 4.07 0.59 0.110 T = 150° C., IsoparE:585 g, 1-octene: 15 g; ΔH₂: 140 psi, Ethylene: 150 psi, Catalyst 7: 0.3μmole, Co-Catalyst A: 0.36 μmole, MMAO-3A: 10 μmole.

Utilizing those polymerization conditions to produce high molecularweight polymers resulted in tetra-functional “Ladder Branching”, whichoccurred when the diene was incorporated into the polymerizationreaction. The polymerization reaction yielded polymer resins with highmolecular weights and tetra-functional “Ladder Branching”.

Example 3: Conditions that Yield Branched Homopolymer

TABLE 15 High density polyethylene examples using decadiene andpentadiene Diene Poly Conventional GPC Data and Metrics Added YieldT_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (° C.) (g/mole) Mwo MpoG_((79/29)) A_(TAIL) 15.C 0 0.00 7.1 133.2 18,988 60,271 50,119 1.001.00 0.12 0.028 15.1 1,9-decadiene 0.15 7.0 135.0 23,754 102,744 58,8841.70 1.17 0.04 0.033 15.2 1,4-pentadiene 0.1 2.3 133.9 22,525 104,21670,795 1.73 1.41 −0.06 0.013 15.3 1,4-pentadiene 0.2 1.3 135.1 28,186114,916 81,283 1.91 1.62 −0.23 0.015 T = 160° C., IsoparE: 600 g,1-octene: 0 g; ΔH₂: 240 psi, Ethylene: 150 psi, Catalyst 7: 0.4 μmole,Co-Catalyst A: 0.48 μmole, MMAO-3A: 10 μmole.

Incorporating diene into the polymerization reaction used to makehomopolymers (with a small amount of diene) resulted in an increase inmolecular weight (tetra-functional “Ladder Branching”). The datarecorded in Table 15 indicated that the examples of ethylene only resinsincreased in molecular weight when two different dienes whereincorporated into the polymerization reaction.

TABLE 16 Lower density polyethylene examples using pentadiene withCatalyst 8 Diene Poly Octene Conventional GPC Data and Metrics AddedYield in Poly T_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.)(g/mole) Mw_(o) Mp_(o) G_((79/29)) A_(TAIL) 16.C none 0 3.4 7.0 79.521,739 65,347 53,703 1.00 1.00 0.12 0.028 16.1 1,4-pentadiene 0.1 1.47.3 81.2 23,203 75,022 57,544 1.15 1.07 0.14 0.031 16.2 1,4-pentadiene0.2 1.9 7.0 81.2 25,701 84,942 60,256 1.30 1.12 0.06 0.024 T = 150° C.,IsoparE: 555 g, 1-octene: 45 g; ΔH₂: 220 psi, Ethylene: 150 psi,Catalyst 8: 0.4 μmole, Co-Catalyst A: 0.48 μmole, MMAO-3A: 10 μmole.

The results in Table 16 indicated that branching occurs with differentcatalysts and at different densities. The resins in Table 16demonstrated branching with Catalyst 8 and enough octene for 7 mol % inthe polymer.

TABLE 1 7 Lower density polyethylene examples using pentadiene anddecadiene at higher linear MW with Catalyst 8 Diene Poly ConventionalGPC Data and Metrics Added Yield Oct. T_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene(g) (g) mol % (° C.) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 17.C none0.00 5.4 — 85.1 31,155 100,236 79,433 1.00 1.00 0.10 0.024 17.11,9-decadiene 0.30 5.6 — 82.0 28,449 143,906 104,713 1.44 1.32 −0.660.021 17.2 1,4-pentadiene 0.20 2.0 — 84.5 33,901 141,038 100,000 1.411.26 −0.76 0.014 T = 150° C., IsoparE: 555 g, 1-octene: 45 g; ΔH₂: 140psi, Ethylene: 150 psi, Catalyst 8: 0.3 μmole, Co-Catalyst A: 0.36μmole, MMAO-3A: 10 μmole.

Based on the results in Table 17, molecular weight increased withaddition of diene indicative of tetra-functional “Ladder Branching”.These examples had higher linear molecular weights. In examples 5.1 and5.2, Catalyst 8 produced polymer resins with a higher molecular weightwhen decadiene or pentadiene was present in the polymerizationreactions.

TABLE 18 Hexene used as a comonomer in place of octene and comparingdifferent catalysts with decadiene Diene Poly Hexene Conventional GPCData and Metrics Added Yield in Poly T_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene(g) (g) (mol %) (° C.) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 18.1.C*none 0 11.5 2.4 114.3 21,627 61,061 47,863 1.00 1.00 0.13 0.029 18.1.1*1,-decadiene 0.35 5.2 2.4 117.1 24,307 132,150 93,325 2.16 1.95 −0.120.036 18.2.C** none 0 7.9 5.6 103.5 17,419 58,044 50,119 1.00 1.00 0.090.026 18.2.1** 1,9-decadiene 0.25 4.0 5.5 92.2 22,482 96,617 70,795 1.661.41 −0.03 0.021 T = 150° C., Ethylene: 150 psi, MMAO-3A: 10 μmole,*IsoparE: 585 g, 1-hexane: 15 g; ΔH₂: 240 psi, Catalyst 7: 0.3 μmole,Co-Catalyst A: 0.36 μmole **IsoparE: 555 g, 1-hexane: 4 5g; ΔH₂: 200psi, Catalyst 8: 0.3 μmole, Co-Catalyst A: 0.36 μmole.

The results in Table 18 indicated that increases in molecular weight(tetra-functional “Ladder Branching”) occurred when a different α-olefincomonomer was used. The polymer resins in Table 18 were produced by twodifferent catalysts and two different loadings of hexene.

TABLE 19 Lower density polyethylene examples using decadiene at higherlinear MW with Catalyst 7 Diene Poly Octene Conventional GPC Data andMetrics Added Yield in Poly T_(m) Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g)(mol %) (° C.) (g/mole) Mwo Mpo G_((79/29)) A_(TAIL) 19.C None 0.00 8.57.1 78.1 37,359 102,362  77,625 1.00 1.00 0.12 0.026 19.1 1,9-decadiene0.15 4.9 5.0 84.2 30,255 159,937  87,096 1.56 1.12 −1.44 0.017 19.21,9-decadiene 0.30 9.0 6.9 80.4 45,031 194,779 109,648 1.90 1.41 −0.260.031 T = 150° C., IsoparE: 555 g, 1-octene: 45 g; ΔH₂: 600 psi,Ethylene: 150 psi, Catalyst 7: 0.3 μmole, Co-Catalyst A: 0.36 μmole,MMAO-3A: 10 μmole.

Based on the results in Table 19, increases in molecular weight withdiene (tetra-functional “Ladder Branching”) occurred with differentlevels of octene. The examples in Table 19 indicated that even with 7mol % octene in the polymer, tetra-functional “Ladder Branching”occurred.

TABLE 20 Branching with different dienes such as pentadiene Diene PolyOctene Conventional GPC Data and Metrics Added Yield in Poly Tm Mn Mw MpMw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.) (g/mole) Mwo Mpo G_((79/29))A_(TAIL) 20.0 none 0.00 6.3 7.5 77.1 29,605  83,716 64,565 1.00 1.00 0.12 0.028 20.1 1,9-decadiene 0.30 7.0 5.0 81.3 35,781 160,761 85,1141.92 1.32 −0.72 0.028 20.2 1,4-pentadiene 0.20 2.3 4.8 85.3 24,397132,720 87,096 1.59 1.35 −0.54 0.012 T = 150° C., IsoparE: 555 g,1-octene: 45 g; ΔH₂: 100 psi, Ethylene: 150 psi, Catalyst 7: 0.3 μmole,Co-Catalyst A: 0.36 μmole, MMAO-3A: 10 μmole.

As evidenced in Table 20, tetra-functional “Ladder Branching” occurredwith different levels of octene and higher starting molecular weights.The examples 8.1 and 8.2 indicated that a polymer resin with 7 mol %octene and starting M_(w) of approximately 83,000 g/mol led to branchingboth with decadiene and pentadiene.

TABLE 21 Branching with high levels of octene and low linear molecularweights. Diene Poly Octene Conventional GPC Data and Metrics Added Yieldin Poly Tm Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.) (g/mole)Mwo Mpo G_((79/29)) A_(TAIL) 21.0 none 0.00 12.8 9.6 69.6 15,073 43,07438,019 1.00 1.00  0.15 0.028 21.1 1,9-decadiene 0.45  4.2 9.0 74.118,788 82,682 50,119 1.92 1.32 −0.03 0.029 T = 150° C., IsoparE: 542 g,1-octene: 58 g; ΔH₂: 200 psi, Ethylene: 150 psi, Catalyst 7: 0.3 μmole,Co-Catalyst A: 0.36 μmole, MMAO-3A: 10 μmole.

The results in Table 21 showed there was an increase in molecular weight(tetra-functional “Ladder Branching”) at much lower density (high levelsof octene in polymer) and lower starting molecular weight. In example9.1, the polymer resin had greater than 9 mol % octene and startingM_(w) of approximately 43,000 g/mol. When diene was incorporated intothe polymerization reaction, molecular weight increased (“LadderBranching” occurred).

TABLE 22 “Ladder Branching” with decadiene at lower linear molecularweights Diene Poly Octene Conventional GPC Data and Metrics Added Yieldin Poly Tm Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.) (g/mole)Mwo Mpo G_((79/29)) A_(TAIL) 22.0 none 0.00 9.5 7.4 81.5 17,778 51,21142,658 1.00 1.00  0.13 0.027 22.1 1,9-decadiene 0.25 8.9 7.7 77.9 18,22870,134 43,652 1.37 1.02 −0.03 0.027 T = 150° C., IsoparE: 555 g,1-octene: 45 g; ΔH₂: 180 psi, Ethylene: 150 psi, Catalyst 7: 0.3 μmole,Co-Catalyst A: 0.36 μmole, MMAO-3A: 10 μmole.

The results in Table 22 indicated that molecular weight incased withdiene (tetra-functional “Ladder Branching”) with a different level ofincorporated octene at a lower starting molecular weight. In example22.1, the starting molecular weight of the polymer resin wasapproximately 51,000 g/mol, and when diene was incorporated into thepolymerization reactions, the molecular weight increased to 70,000 g/mol(tetra-functional “Ladder Branching” occurred).

TABLE 23 Different ethylene pressure and octene added to the reactorDiene Poly Octene Conventional GPC Data and Metrics Added Yield in PolyTm Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.) (g/mole) Mwo MpoG_((79/29)) A_(TAIL) 23.0 none 0.00 10.0 6.6 83.4 28,232  82,570 66,0691.00 1.00  0.10 0.026 23.1 1,9-decadiene 0.38  9.5 6.6 83.1 30,603119,347 70,795 1.45 1.07 −0.20 0.014 T = 150° C., IsoparE: 533 g,1-octene: 67 g; ΔH₂: 240 psi, Ethylene: 233 psi, Catalyst 7: 0.3 μmole,Co-Catalyst A: 0.36 μmole, MMAO-3A: 10 μmole.

According to the data in Table 23 and Table 24, molecular weightsincreased (tetra-functional “Ladder Branching” occurred) when theethylene pressure and amount of octene in the reactor was increased.

TABLE 24 Varied ethylene pressure and amount of octene added to thereactor. Diene Poly Octene Conventional GPC Data and Metrics Added Yieldin Poly Tm Mn Mw Mp Mw/ Mp/ Ex. Diene (g) (g) (mol %) (° C.) (g/mole)Mwo Mpo G_((79/29)) A_(TAIL) 24.0 none 0.00 10.7 6.0 90.6 47,020 128,732102,329 1.00 1.00 −0.01 0.016 24.1 1,9-decadiene 0.50 10.4 5.9 88.746,782 182,025 112,202 1.41 1.10 −1.07 0.018 T = 150° C., IsoparE: 510g, 1-octene: 90 g; ΔH₂: 240 psi, Ethylene: 300 psi, Catalyst 7: 0.3μmole, Co-Catalyst A: 0.36 μmole, MMAO-3A: 10 μmole.

Example 4: Catalyst 7 Catalyzed Decadiene-Homopolymer

TABLE 25 No octene added to the reactor. Diene Poly Conventional GPCData and Metrics Added Yield Tm Mn Mw Mp Ex. Diene (g) (g) (° C.)(g/mole) G_((79/29)) A_(TAIL) 25.C.1 none 0.00  8.1 133.1  24,817 71,529  63,096  0.07 0.023 25.C.2 none 0.00  8.4 134.9  23,596  68,706 60,256  0.06 0.022 25.1 1,9-decadiene 0.03  8.3 138  27,636  84,420 63,096  0.11 0.027 25.2 1,9-decadiene 0.06  8.5 137.7  26,305  94,520 67,608 −0.01 0.023 25.3 1,9-decadiene 0.10  8.6 138.5  27,549 106,799 75,858 −0.06 0.019 25.4 1,9-decadiene 0.20  8.2 138.7  31,246 149,195120,226  0.03 0.038 Diene Absolute GPC Data and Metrics Added Mn Mw MpEx. Diene (g) (g/mole) G_((79/29)) A_(TAIL) gpcBR 25.C.1 none 0.0025.C.2 none 0.00 18,771  60,641  47,865  0.06  0.027  0.07 25.11,9-decadiene 0.03 21,119  81,512  52,483  0.12  0.035  0.17 25.21,9-decadiene 0.06 26,147  97,809  51,288  0.07  0.035  0.25 25.31,9-decadiene 0.10 20,182 116,859  66,072  0.04  0.031  0.34 25.41,9-decadiene 0.20 29,735 242,736 109,653  0.14  0.075  0.92 T = 150°C., IsoparE: 600 g, 1-Octene: 0 g (except Entry 25.C.1 which has 0.2 g),ΔH₂: 240 psi, Ethylene: 150 psi, Catalyst 7: 0.4 μmole, Co-Catalyst A:0.48 μmole, MMAO-3A: 10 μmole.

Results summarized in Table 25 indicate that tetra-functional “LadderBranching” occurs when octene is not present in the reactor. Themolecular weight of each example in Table 25 increased as the amount ofdecadiene in the polymerization reaction increased. PGP-15 T

TABLE 26 Carbon and proton NMR evaluation of methines, vinyls, andvinylenes (per 1000 carbon atoms) of the Examples recorded in Table 25.Octene Decadiene methines Vinyls Vinylenes Example (g) (g) per 1000 Cper 1000 C per 1000 C 26.C.1 0.2 0.00 0.38 0.04 0 26.C.2 0.0 0.00 00.041 0 26.1 0.0 0.03 0.08 0.056 0 26.2 0.0 0.06 0.22 0.085 0.002 26.30.0 0.10 0.3 0.12 0.003 26.4 0.0 0.20 0.45 0.2 0.005

Example 26.C.1 is the result of a polymerization reaction containing 0.2octene. FIG. 29 is a graph of the Log(MW) of the Examples 26.C.1,26.C.2, and 26.1-26.4. As the amount of decadiene increases, themolecular weight peak shifts right. Table 27 to Table 32 summarize theresults of a Dynamic Mechanical Spectrum for Examples 26.C.1, 26.C.2,and 26.1-26.4. The results of each Table 27 to Table 32 indicate theelasticity factor, m, decreases as the amount of tetra-functional“Ladder Branching” increases. Additionally, the results of each Table 27to Table 32 indicate the rheology ratio increases as the amount oftetra-functional “Ladder Branching” increases.

FIG. 29 is a conventional molecular weight distribution of curve ofseries 26.C.1, 26.C.2, and 26.1 to 26.4.

TABLE 27 Dynamic Mechanical Spectrum of Example 26.C.1, a linear shortchain branched polymer, at 190° C.. Ang Storage Loss Complex Complexfreq modulus modulus viscosity Tan modulus Phase rad/s Pa Pa Pa.s(delta) Pa angle ° 0.10 0 76 762 192.81 76 89.7 0.16 1 121 761 145.80121 89.6 0.25 1 191 760 131.04 191 89.6 0.40 4 302 758 74.41 302 89.20.63 9 477 755 55.57 477 89.0 1.00 18 753 753 42.51 753 88.7 1.58 381186 749 31.35 1187 88.2 2.51 81 1865 743 22.98 1866 87.5 3.98 170 2922735 17.22 2927 86.7 6.31 351 4561 725 13.00 4575 85.6 10.00 718 7078 7119.86 7115 84.2 15.85 1448 10894 693 7.52 10989 82.4 25.12 2865 16568 6695.78 16814 80.2 39.81 5513 24807 638 4.50 25412 77.5 63.10 10265 36398599 3.55 37818 74.3 100.00 18369 52036 552 2.83 55183 70.6

The Dynamic Mechanical Spectrum of the comparative was measured and theresults recorded in Table 27. The shear viscosity at 0.1 radians/secondwas calculated to be 762 Pa s and the shear viscosity at 100radians/second was measured at 552 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 1.4. The tan (δ_(0.1)) of the branched polymer inExample 26.C.1 was 192.8, and the tan (δ₁₀₀) was 2.8, which yields anelasticity factor of 1901.6 at 190° C.

TABLE 28 Dynamic Mechanical Spectrum of Example 26.C.2, a linearpolymer, at 190° C.. Ang Storage Loss Complex Complex freq modulusmodulus viscosity Tan modulus Phase rad/s Pa Pa Pa.s (delta) Pa angle °0.10 0 66 662 401.31 66 89.9 0.16 0 105 662 212.34 105 89.7 0.25 1 166661 126.46 166 89.5 0.40 3 263 661 91.50 263 89.4 0.63 6 416 659 70.63416 89.2 1.00 13 658 658 51.74 658 88.9 1.58 28 1038 655 37.74 1039 88.52.51 60 1635 651 27.25 1636 87.9 3.98 127 2567 646 20.19 2570 87.2 6.31267 4018 638 15.07 4027 86.2 10.00 553 6256 628 11.31 6281 84.9 15.851131 9671 614 8.55 9737 83.3 25.12 2272 14793 596 6.51 14967 81.3 39.814439 22291 571 5.02 22729 78.7 63.10 8445 32969 539 3.90 34034 75.6100.00 15436 47687 501 3.09 50123 72.1

The Dynamic Mechanical Spectrum of the comparative, Example 26.C.1, wasmeasured and the results recorded in Table 28. The shear viscosity at0.1 radians/second was calculated to be 662 Pa s and the shear viscosityat 100 radians/second was measured at 501 Pa s, providing a rheologyratio (V_(0.1)/V₁₀₀) of 1.3. The tan (δ_(0.1)) of the linear polymer ofExample 26.C.1 was 401.3, and the tan (δ₁₀₀) was 3.1, which yields anelasticity factor of 3986.2 at 190° C.

TABLE 29 Dynamic Mechanical Spectrum of Example 26.1, a tetra-functional “Ladder Branched” polymer, at 190° C.. Ang Storage LossComplex Complex freq modulus modulus viscosity Tan modulus Phase rad/sPa Pa Pa.s (delta) Pa angle° 0.10 168 722 7410 4.29 741 76.9 0.16 2931054 6900 3.60 1094 74.5 0.25 492 1514 6336 3.08 1591 72.0 0.40 797 21385731 2.68 2282 69.6 0.63 1250 2973 5112 2.38 3225 67.2 1.00 1900 40824502 2.15 4502 65.0 1.58 2810 5540 3919 1.97 6211 63.1 2.51 4059 74493377 1.84 8483 61.4 3.98 5741 9964 2888 1.74 11499 60.0 6.31 7995 132872458 1.66 15507 59.0 10.00 11019 17695 2085 1.61 20845 58.1 15.85 1510023534 1764 1.56 27961 57.3 25.12 20749 31208 1492 1.50 37476 56.4 39.8128424 41229 1258 1.45 50078 55.4 63.10 39014 54069 1057 1.39 66675 54.2100.00 53608 70156 883 1.31 88294 52.6

The Dynamic Mechanical Spectrum of Example 26.1 was measured and theresults recorded in Table 29. The shear viscosity at 0.1 radians/secondwas calculated to be 7,410 Pa s and the shear viscosity at 100radians/second was measured at 883 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 8.4. The tan (δ_(0.1)) of the branched polymer inExample 13.1 was 4.3, and the tan (δ₁₀₀) was 1.3, which yields anelasticity factor of 29.8 at 190° C.

TABLE 30 Dynamic Mechanical Spectrum of Example 26.2, a tetra-functional “Ladder Branched” polymer, at 190° C.. Ang Storage LossComplex Complex freq modulus modulus viscosity Tan modulus Phase rad/sPa Pa Pa.s (delta) Pa angle ° 0.10 3624 4341 56549 1.20 5655 50.1 0.164876 5401 45910 1.11 7276 47.9 0.25 6427 6657 36837 1.04 9253 46.0 0.408332 8102 29193 0.97 11622 44.2 0.63 10654 9784 22926 0.92 14465 42.61.00 13448 11749 17858 0.87 17858 41.1 1.58 16769 14037 13798 0.84 2186939.9 2.51 20720 16721 10600 0.81 26626 38.9 3.98 25363 19928 8102 0.7932255 38.2 6.31 30848 23813 6176 0.77 38969 37.7 10.00 37319 28572 47000.77 47001 37.4 15.85 45028 34475 3578 0.77 56711 37.4 25.12 54324 417652728 0.77 68524 37.6 39.81 65674 50951 2088 0.78 83121 37.8 63.10 7971062296 1603 0.78 101170 38.0 100.00 97327 76112 1236 0.78 123550 38.0

The Dynamic Mechanical Spectrum of Example 26.2 was measured and theresults recorded in Table 30. The shear viscosity at 0.1 radians/secondwas calculated to be 56,549 Pa s and the shear viscosity at 100radians/second was measured at 1,236 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 45.8. The tan (60.1) of the branched polymer inExample 26.2 was 1.2, and the tan (δ₁₀₀) was 0.8, which yields anelasticity factor of 4.2 at 190° C.

TABLE 31 Dynamic Mechanical Spectrum of Example 26.3, a tetra-functional “Ladder Branched” polymer, at 190° C.. Ang Storage LossComplex Complex freq modulus modulus viscosity Tan modulus Phase rad/sPa Pa Pa.s (delta) Pa angle ° 0.10 17977 12228 217410 0.68 21741 34.20.16 21869 13969 163730 0.64 25950 32.6 0.25 26179 15829 121790 0.6030592 31.2 0.40 31006 17912 89945 0.58 35808 30.0 0.63 36358 20148 658800.55 41567 29.0 1.00 42361 22662 48042 0.53 48042 28.1 1.58 49016 2543734844 0.52 55223 27.4 2.51 56469 28527 25187 0.51 63266 26.8 3.98 6474632041 18146 0.49 72241 26.3 6.31 73992 36161 13052 0.49 82355 26.0 10.0084312 41035 9377 0.49 93767 26.0 15.85 95926 46900 6737 0.49 106780 26.125.12 109110 54015 4847 0.50 121750 26.3 39.81 124270 62740 3497 0.50139210 26.8 63.10 142000 73445 2534 0.52 159870 27.3 100.00 163090 863541845 0.53 184540 27.9

The Dynamic Mechanical Spectrum Example 26.3 was measured and theresults recorded in Table 31. The shear viscosity at 0.1 radians/secondwas calculated to be 56,549 Pa s and the shear viscosity at 100radians/second was measured at 1,236 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 117.8. The tan (δ_(0.1)) of the branched polymer inExample 26.3 was 1.2, and the tan (δ₁₀₀) was 0.8, which yields anelasticity factor of 4.2 at 190° C.

TABLE 32 Dynamic Mechanical Spectrum of Example 26.4, a tetra-functional “Ladder Branched” polymer, at 190° C.. Ang Storage LossComplex Complex freq modulus modulus viscosity Tan modulus Phase rad/sPa Pa Pa.s (delta) Pa angle ° 0.10 86332 28452 909000 0.33 90900 18.20.16 95178 30306 630240 0.32 99887 17.7 0.25 104510 32167 435310 0.31109340 17.1 0.40 114280 34156 299610 0.30 119280 16.6 0.63 124490 36260205500 0.29 129660 16.2 1.00 135390 38643 140790 0.29 140790 15.9 1.58146880 41177 96245 0.28 152540 15.7 2.51 159050 43788 65676 0.28 16497015.4 3.98 172020 46770 44777 0.27 178260 15.2 6.31 185770 50056 304930.27 192400 15.1 10.00 200450 53794 20754 0.27 207540 15.0 15.85 21610058080 14119 0.27 223770 15.0 25.12 232940 63088 9607 0.27 241330 15.239.81 251110 69110 6542 0.28 260440 15.4 63.10 270990 76373 4462 0.28281550 15.7 100.00 293290 85175 3054 0.29 305410 16.2

The Dynamic Mechanical Spectrum of Example 26.4 was measured and theresults recorded in Table 32. The shear viscosity at 0.1 radians/secondwas calculated to be 909,000 Pa s and the shear viscosity at 100radians/second was measured at 3,054 Pa s, providing a rheology ratio(V_(0.1)/V₁₀₀) of 297.6. The tan (δ_(0.1)) of the branched polymer inExample 26.4 was 0.3, and the tan (δ₁₀₀) was 0.3, which yields anelasticity factor of 0.4 at 190° C.

Guzman-2010 demonstrated and analyzed the MWD and physical propertiesresulting from conventional diene branching in a steady-state CSTR. Aconstrained geometry catalyst (CGC) was used to copolymerize ethylene,1-octene, and 1,9-decadiene in a very well mixed one-gallon reactorsystem. The particular CGC catalyst, used by Guzman, was described indetail by U.S. Pat. No. 5,965,756 (structure IX) and U.S. Pat. No.7,553,917 (Example 3). The Guzman-2010 catalyst was designed to grow asingle chain from the catalyst center. Guzman's data were gathered atsteady state while operating a CSTR at a pressure of 525 psig and atemperature of 155° C. over a range of diene feed concentrations. Thevarious steady-state polymer samples collected by Guzman contained nomeasurable levels of gels or insoluble material. However, at the highestlevel of dienes feed some minor internal reactor fouling was observed,and it was anticipated that higher levels of dienes feed would result ingels formation or reactor MWD instability.

In Table 33, a selected series of data from Guzman was summarized forotherwise fixed reactor conditions over a spectrum of diene feed levels.Throughout the series, the ethylene and 1-octene feed concentrationswere set at 13.8 wt % and 3.6 wt %, respectively. The catalyst feed ratewas continuously adjusted to maintain a constant ethylene conversion of79% throughout the series resulting in a fixed polymer production rateof 2.2 kg/hr. The polymer density, a measure of copolymer composition,was constant at about 0.922 g/cc.

TABLE 33 Comparative example of Guzman CS TR results using asingle-chain constrained geometry catalyst and 1,9-decadiene.1,9-decadiene Conventional GPC Data Absolute GPC Data Feed Incorp.vinyls/ I₂ M_(n) M_(w) M_(p) M_(n) M_(w) M_(p) Sample (ppm)^(A)(ppm)^(B) 1000C (dg/min) I₁₀/I₂ (kg/mole) (kg/mole) 33.C1  0   0 0.11 20 6.2 19.9 42.5 35.8 21.5 48.2 39.1 33.C2 523 2119 0.25  7.2  9.1 19.452.9 36.5 20.2 58.9 38.6 33.C3 704 2912 0.29  4.6 10.2 19.5 55.5 37.421.4 67.2 39.1 33.C4 794 3186 0.30  3.2 11.0 20.9 59.5 36.4 22.5 73.639.0 33.C5 837 3405 0.30  3.1 11.5 20.2 60.2 36.9 21.7 75.3 38.5 33.C6881 3502 0.28  2.1 11.8 21.6 65.2 38.9 23.0 81.5 41.9 33.C7 923 39460.31  1.3 13.0 21.9 70.4 40.8 23.9 90.4 39.9 ^(A)1,9-decadiene feedlevel expressed as overall mass fraction, in units of ppm^(B)1,9-decadiene incorporation as expressed in polymer mass fraction,in units of ppm.

The data in Table 33 demonstrated how changes in conventional dienebranching level affects average molecular weight and polydispersity aswell as properties such as viscosity, as reflected by I₂ and I₁₀. Theeffect of conventional diene branching on molecular weight was shown inTable 33 for both absolute and conventional MWD measurement techniques.While absolute MWD measurement is the preferred method for branchedpolymers, it is not always available. Therefore, Table 33 also containsmolecular weights measured by conventional techniques using a refractiveindex detector. The results in Table 33 demonstrated that, by eithermeasurement technique, the weight average molecular weight (M_(w)) risessubstantially as the diene feed is increased from zero to 923 ppm.

Though not reported in Guzman, the MWD curves associated with Table 33were found and plotted in FIGS. 30A and 30B for absolute andconventional GPC measurement techniques, respectively. The MWD curvedata in FIGS. 30A and 30B demonstrated that the expected high M_(w) tailformation resulting from conventional diene branching occurred. The lackof significant movement of the peak MW with increasing diene branchingis also apparent from the MWD curves.

The molecular weight distributional data in FIGS. 30A and 30B werereduced to simple metrics describing the evolution of the MWD curveposition and shape as more diene monomers were fed to the CSTR. The datain Table 34 showed these MWD metrics for both absolute and conventionalMWD measurements of the Guzman's polymer samples. Absolute MWDmeasurement data in Table 34 showed up to an 87% increase in molecularweight as 1,9-decadiene feed ranged from 0 to 923 ppm. Peak molecularweight change, as indicated by M_(p), does not vary significantly foreither means of molecular weight measurement, which is inconsistent with“Ladder branched” polymer results. The shape factors are summarized inTable 34 and are inconsistent with “Ladder branched” polymers becausethe values for both G_(79/29) and A_(TAIL) increased as the diene feedlevel and M_(w) increased.

TABLE 34 Molecular weight data and metrics associated with examples inTable 33. Conventional GPC Metrics Absolute GPC Metrics Diene M_(w)/M_(p)/ M_(w)/ M_(p)/ Ex. Sample Feed (ppm) M_(w0) M_(p0) G_((79/29))A_(TAIL) M_(w0) M_(p0) G_((79/29)) A_(TAIL) 33.C1 19  0 1.00 1.00 0.120.011 1.00 1.00 0.06 0.026 33.C2 20 523 1.24 1.02 0.22 0.043 1.22 0.990.18 0.062 33.C3 21 704 1.31 1.05 0.23 0.047 1.39 1.00 0.15 0.058 33.C422 794 1.40 1.02 0.27 0.051 1.53 1.00 0.13 0.062 33.C5 23 837 1.42 1.030.29 0.055 1.56 0.98 0.17 0.062 33.C6 24 881 1.53 1.09 0.26 0.061 1.691.07 0.20 0.066 33.C7 25 923 1.66 1.14 0.33 0.063 1.87 1.02 0.23 0.071

Some key parameters on commercial resins are tabulated in Table 35. Someof the basic parameters for materials were made in solution, gas phase,and high pressure reactors.

TABLE 35 Physical Characteristics of non-“Ladder Branched” PolymerCompositions Rheology Average Ratio Example Source Data Resin c-PDI g′V_(0.1) V₁₀₀ (RR) Tanδ_(0.1) Tanδ₁₀₀ 33.C1 DOWLEX LLDPE 4.35 0.91 8,5251,612 5.3 9.61 0.97 2045G 33.C2 ASPUNE 6835A LLDPE 3.27 0.99 1,064 5032.1 25.2 1.96 33.C3 ATTANE 4201 ULDPE 4.59 0.86 9,643 1,686 5.7 8.160.93 33.C4 ELITE ™ 5800G LDPE 966 304 3 33.C5 Affinity ™ PL LDPE 9,9801,404 7.1 1880 33.C6 LDPE 722 LDPE 10.31 0.606 1987 254 7.8 6.01 1.0133.C7 LDPE 5004I LDPE 8.08 0.624 3420 335 10.2 4.73 0.922 33.C8 LDPE662I LDPE 11.55 0.548 21,078 646 32.6 1.63 0.70 33.C9 Exxon Resin 1*LDPE 4.43 0.63 28,266 913 31.0 2.09 0.65 33.C10 Equistar Resin* LDPE7.67 0.54 37,944 730 52.0 1.26 0.59 33.C11 Exxon Resin 2* LDPE 4.69 0.5716,153 751 21.5 2.86 0.73 33.C8 US 9580533 Ethylene- 14.15 0.979 9283384.4 24.1 — — (Slurry) diene 33.C12 US 9580533 Ethylene- 7.45 0.51320162 521.5 38.7 — — (Solution) diene 33.C13 US 9580533 Ethylene- 3.680.880 173.8 134.2 1.3 — — (Solution) diene 33.C14 US 9580533 Ethylene-5.28 0.718 16847.5 842.8 20.0 — — (Solution) diene 33.C15 US 9580533Ethylene- 7.97 0.696 9893.2 569.6 17.4 — — (Solution) diene 33.C16 US9580533 Ethylene- 5.71 0.593 1214.2 352.6 3.4 — — (Solution) diene ™Trademark of The Dow Chemical Company*Competitor resins were tested as comparisons. The Exxon Resins wereobtained from ExxonMobil. Equistar Resin is a LyondellBasell Petrotheneproduct.

The data summarized in Table 35 are plotted in the graphs of FIG. 31 andFIG. 32. The data illustrates the difference in the “Ladder Branched”polymers in comparison to LDPE, LLDPE, ULDPE, and ethylene resinscontaining diene monomers. In FIG. 31 and FIG. 32, the “Ladder Branched”polymers (Ladder-PE in the legend of the graph) of this disclosure areclustered together, thus indicating that the “Ladder Branched” polymershave unique polymeric properties in comparison to other ethylene-basedresins. As shown in the graph of FIG. 31, the “Ladder Branched” polymershave a rheology ratio at least 10, and an average g′ less than 0.86. InFIG. 31, the LDPE resins plotted have an average g′ of less than 0.65;the prior art ethylene-diene resins (listed as Prior Art ET-Diene in thelegend) do not cluster together.

In FIG. 33, the melt strength (centiNewtons, cN) was measured as afunction of the melt index (Log I₂). The polymers produced from the dualchain catalyst, as indicated by the triangles and the circles, werecompared to a polymer produced from a single chain catalyst, andliterature based curves of autoclave LDPE, tubular LDPE, and linearpolyethylene. The melt strengths of the polymers produced from the dualchain catalysts were less than the melt strength of the autoclave LDPE,tubular LDPE, and the polymer produced from the single chain catalysts,but significantly greater than the linear polyethylene. This wouldindicate that the polymers produced from the dual chain catalysts haveentangled, long-chain branching.

It should be apparent to those skilled in the art that variousmodifications can be made to the described embodiments without departingfrom the spirit and scope of the claimed subject matter. Thus, it isintended that the specification cover modifications and variations ofthe described embodiments provided such modification and variations comewithin the scope of the appended claims and their equivalents.

1. A process of synthesizing long-chain branched copolymers, the processcomprising: contacting together one or more C₂-C₁₄ alkene monomers, atleast one diene or polyene, optionally a solvent, and a multi-chaincatalyst, wherein the multi-chain catalyst comprises multiplepolymerization sites; producing at least two polymer chains of theC₂-C₁₄ alkene monomers, each polymer chain polymerizing at one of thepolymerization sites; and synthesizing the long-chain branched polymersby connecting the two polymer chains with the diene or polyene, theconnecting of the two polymer chains being performed in a concertedmanner during the polymerization.
 2. The process of claim 1, whereinethylene is added to such an extent that the resulting copolymer contentis greater than 50 mol % ethylene.
 3. The process of claim 1, whereinthe diene and polyene comonomer is incorporated at such level to achieveat least one bridging juncture per 100 copolymer chains.
 4. The processof claim 1, wherein the diene is unconjugated or the polyene has atleast two unconjugated bonds per molecule.
 5. The process of claim 1,wherein the multi-chain catalyst is a heterogeneous catalyst withsurface concentration of metal atoms greater than or equal to 0.3 metalatoms per nanometer squared (metal/nm²).
 6. The process of claim 1,wherein the multi-chain catalyst has two ligated transition metalslinked by a dianionic activator where the distance between the metalatoms is less than or equal to 18.5 Å.
 7. The process of claim 1,wherein the multi-chain catalyst consists of two or more transitionmetals covalently tethered, where the distance between the metal atomsis less than or equal to 18.5 Å.
 8. The process of claim 1, wherein themulti-chain catalyst has two or more polymer chains on the same metal.9. The process of claim 1, wherein the multi-chain catalyst has amonoanionic ligand, is a Group IV Metal (Ti, Zr, Hf, and has two polymerchains on the same metal.
 10. The process of claim 1, wherein thelong-chain branched copolymer M_(w) is at least 20% greater than theM_(w) of the polymer synthesized without the diene or polyene or whereinthe long-chain branched copolymer M_(p) is at least 20% greater than theM_(p) of the polymer synthesized without the diene or polyene.
 11. Theprocess of claim 1, wherein the polydisperse long-chain branched polymerhas from 0.01 to 0.5 diene junctures per number average copolymer chainor 0.02 to 1.0 diene junctures per weight average copolymer chain. 12.The process of claim 1, wherein the polymerization occurs in a solutionpolymerization reactor or a particle forming polymerization reactor suchas a slurry reactor or a gas phase reactor, wherein the molecular orsolid-supported catalyst is delivered to the reaction media or developedin the reaction media, wherein the reactor system is batch or continuousor a hybrid such as semi-batch, wherein the reactor residence timedistribution is narrow such as in non-backmixed reactors or broad suchas in backmixed reactor and series and recycle reactors.
 13. The processof claim 1, wherein the diene is linear.
 14. The process of any one ofclaim 1, wherein the diene is selected from 2-methyl-1,4-pentadiene,3-methyl-1,4-pentadiene, 1,3-divinylcyclopentane,2-methyl-1,5-hexadiene, 1,4-pentadiene, 1,5-hexadiene, 1,7-octadiene,1,8-nonadiene, 1,9-decadiene, 1,11-dodecadiene, and 1,15-hexadecadiene.